In this answer, I’ll discuss the general problem of dividing a line into feet, and then I’ll look in more detail at Chaucer’s opening line.
Are feet real?
English prosody is based on stress, which is a real feature of the spoken language, manifesting as variation in volume, pitch, and vowel reduction. So, marking the stressed syllables in bold, you might read the line as
Whan that Aprill with his shoures soote
But what about feet? There does not seem to be any feature of spoken language that corresponds to foot divisions: no-one pauses or makes a sound to mark a foot division. Because of this, many prosodists who have looked at the question have concluded that foot divisions are arbitrary in English verse, for example C. M. Lewis:
Feet are arbitrary and phantom concepts, and this description of the line does not touch the really vital fact about its rhythm,—the underlying regularity of its time-scheme. […] there is much dispute about the scansion of the line quoted. It seems to me, however, that all such disputes are teapot tempests. Since feet are merely matters of convenience, and rival systems of scansion merely different devices for attaining convenience at the expense of scientific accuracy, it seems useless to spin theories about them.
C. M. Lewis (1907). The Principles of English Verse, p. 43. New York: Henry Holt.
So you could scan the line as trochaic pentameter:
Whan that | Aprill | with his | shoures | soote
or as headless iambic pentameter with a hypercatalectic (or “feminine”) ending:
Whan | that Ap|rill with | his shour|es soot|e
Given this line on its own, the trochaic scansion seems preferable because it is simpler. But lines appear in a context, and the context of the Canterbury Tales is that the majority of the lines are perfect iambic pentameter with an optional hypercatalectic ending. For example, the General Prologue continues:
The droghte of March hath perced to the roote
And bathed every veyne in swich licour,
Of which vertu engendred is the flour;
Whan Zephirus eek with his sweete breeth
Inspired hath in every holt and heeth
The tendre croppes, and the yonge sonne
Hath in the Ram his halfe cours yronne
where the only deviations from this rhythm are the two unstressed syllables in “every”, but most readers would pronounce the word as “ev’ry” to make it fit. In this strongly iambic context it is simpler to treat the first line as iambic too, but with an initial omitted syllable. But both descriptions really amount to the same thing.
See this answer for more about the arbitrariness of feet in English verse, and this answer for more about the way that scansion depends on the context.
What did Chaucer write?
The version of the opening line that you quoted from the blog is common in modern editions of Chaucer, for example it appears in The Riverside Chaucer, but it does not correspond to any of the earliest manuscripts, and it is an editorial reconstruction.
Let’s take a look at three of the oldest surviving manuscripts to see how they spelled the line. Click for larger screenshots, or follow the links to the complete digitized manuscripts.
The National Library of Wales’ Peniarth 392D or Hengwrt Chaucer (c. 1395–c. 1405) has:
Whan that Aueryll wt his shoures soote

The Ellesmere Manuscript (c. 1400–c. 1405) has:
Whan that Aprill with hise shoures soote

The British Library’s Harley MS 7334 (c. 1410) has:
Whan that aprille with his schowres swoote

You’ll see that in each of these manuscripts there seems to be an extra syllable compared to the version you quoted: in the Peniarth manuscript “Aueryll” has three syllables, in the Harley manuscript “aprille” has three syllables (if the “e” is pronounced), and in the Ellesmere manuscript, “hise” has two (ditto).
How can we explain this variety of forms for the first line? One way to explain it is that Chaucer’s original line was difficult in some way, and so the scribes who wrote these manuscripts tried to “correct” the line, but as they were working independently, they ended up “correcting” it in different ways. If Chaucer’s original line had “failed in a syllable”, then the scribes might have tried to add a syllable to regularize it. Perhaps the Hengwrt scribe thought the line might scan
Whan that Aueryll wt his shoures soote
and the Ellesmere scribe thought it might scan
Whan that Aprill with hise shoures soote
and the Harley scribe thought it might scan
Whan that aprille with his schowres swoote
But all of these are unlikely, because Chaucer’s pronunciation of “April” surely had two syllables with the stress on the first syllable, because of this line from the Man at Law’s Prologue:
Of Aprill, that is messager to May
The trochaic reconstruction of the first line originates with John Manly and Edith Rickert:
This line obviously begins with a falling or trochaic rhythm, which is continued throughout the line. It is a mistake to read “Aprille” as trisyllabic with accent on the second syllable. It is true that several MSS spell the word with a final e or with a crossed ll (sometimes supposed to imply a final e), but we have sought in vain for examples of this pronunciation; and several years ago Dr. Robert Bridges told Mr. Manly that at his request Dr. Henry Bradley had made extended search for such evidence but could find none. The word should then be pronounced as a dissyllable with stress on the first syllable. In opening his poem with a trochaic movement Chaucer secured a dash and rapidity which are lacking in the usual manner of reading the line. Such lines are more frequent in CT than has been supposed.
John M. Manly & Edith Rickert (1940). The Text of the Canterbury Tales, volume 3, p. 421. University of Chicago Press.