Why is Fra Pandolf in the plot?

Browning associates "My Last Duchess" strongly with the fifth duke, Alfonso II d'Este, of the Renaissance duchy of Ferrara by adding the single word "Ferrara" as the poem's epigraph, yet he never specifically states Ferrara is the speaker. This suggestive yet evasive association allows us to think of unstated details from a specific time and incident, such as the Duchess's age, while universalizing the themes that Browning illustrates in the narrative of this dramatic monologue, for example, Arrogance and Insolence; Young Brides; Ironic Conflict.
The Duke, in his monologue, implies that the envoy, agent for an unseen Count, who is accompanying the Duke is looking puzzled and wondering about the expression on the Duchess's face: the "earnest glance" of her eyes and the "spot of joy" in the blush of her cheeks (there is no thought here of her smile, just her eyes and cheeks).
The Duke is entertaining the Count's envoy (a person of high station--not a servant--who negotiates contracts or otherwise represents another's interests) by showing him around the palace. This also has a practical function in that the more the envoy knows about the Count's daughter's future home, the better his advice to the Count can be. At the end of the poem, the Duke reiterates his confidence in the Count; his opinions on the daughter's dowry (which will become the Duke's property); his objective in requesting the marriage.
During the Renaissance, friars and monks, if they had the talent, would engage in the art of painting following the example of the renowned Renaissance painter Fra Angelico. That Browning added an allusion to Fra Angelico through the fictional character of Fra Pandolf develops his theme of Arrogance and Insolence by contributing irony and conflict: the Duke is religious and contributes to a religious monastery (money paid to Pandolf will be collected in the monastery's coffers) yet can give "commands" for his wives removal. Browning clarified for Hiram Corson (see Themes Insight) that by "commands" he meant commands for her murder.
"I said / 'Fra Pandolf' by design": Browning also includes Fra Pandolf to help develop the Duchess's character as well as the Duke's character. The story of the marriage of Alfonso II of Ferrara is that at twenty-five, he married a fourteen-year-old girl. In Browning's poem, Fra Pandolf was painting the Duchess's traditional wedding portrait. Knowledge of who the painter was proves that the girl was an innocent and misjudged by the Duke, which is confirmed by the items the Duke accuses her with, like "cherries" and sunset. Knowledge of the painter also shows that the Duke is guilty of uncomprehending arrogance and misjudged his bride to a grievous extent.
While it is tempting to think that the Duke occupied himself with watching the day-long painting process, he does not give quotations of what Fra Pandolf said to the Duchess, rather he makes suppositions of what he perchance said: Something as inconsequential as "your cloak is on your wrist." While denouncing his bride for being "Too easily impressed," he denounces himself for unseeing arrogance and folly. Browning's monologue allows the Duke to expose himself as arrogant and insolent through irony and ironic conflict while exonerating the Duchess as a sweet, simple girl who is not jaded and who still enjoys the simple pleasures of life through a still innocent spirit.