Which aspects of the prioress would Chaucer most likely find disagreeable?

In order to understand Chaucer's attitude toward Prioress Madame Eglentyne as he depicts her in the General Prologue to The Canterbury Tales (ca. 1380-1392 CE), we need to understand the Tales is both an example of satire and the frame narrative. The frame is the pilgrims' journey to visit the tomb of Thomas Becket in Canterbury. More importantly, the poem is also an example of Medieval Estates Satire; that is, satire aimed at the three estates, roughly equivalent to social classes: 1) the Church and clergy; 2) the nobility (those who fought to protect society); and 3), everyone else, which included the peasantry and, later, the middle classes.
As your question implies, Chaucer finds the representatives of the First Estate—beginning with the Prioress—to be less-than-perfect representatives of their class. Chaucer's opening description of the Prioress begins his damnation of her by faint praise:

Ther was also a Nonne, a Prioresse
That of hir smylyng was ful symple and coy;
. . . And she was cleped Madame Eglentyne.
(There was also a nun, a prioresse, who smiled simply and quietly, and her name was Madame Eglentyne)

This begins Chaucer's description of a woman, the head of a convent, who seems to be part of the Second Estate, the nobility, rather than the Church. Rather than taking a name typical of nuns, the Prioress has a name one associates with women of the nobility, and her title—Madame—is not appropriate for a Prioress.
Madame Eglentyne also speaks French, although Chaucer makes it clear she speaks dialectal French (French as it is learned and spoken in England) rather than the preferred French of Paris, poking fun at Madame Eglentyne's pretensions to nobility. By this time, when English had become the language of common speech, speaking French was an affectation, not a necessity. Chaucer's portrait is of a church representative who seems to be confused as to which estate she represents.
In addition to having impeccable table manners, Madame Eglentyne travels with lap dogs:

Of smal houndes hadde she that she fedde
With roasted flessh, or milk and wastrel-breed.
But soore wepte she if oon of hem were deed,
Or if men smoot it with yerd smerte (ll 146-148).
(She had lap dogs that she fed with roasted meat or milk and white bread. And would cry bitterly if one should die or if a man struck one with a stick.)

Chaucer points out that the Prioress is soft-hearted when it comes to her dogs, the implication being that she lavishes inappropriate attention on the dogs to the exclusion of people, who should be her real concern as a Prioress. The fact that she cannot stand to see one of them hurt is yet another example of her refined sensibility, an attribute we would associate with a woman of the noble class.  
Although Chaucer couches his satire in seeming praise of Madame Eglentyne's table manners, personal grooming, speech, and sensibilities, we have the portrait of an important representative of the clergy who is much closer to the nobility than to the church. She is, like the Friar and the Monk, a fraud.