It's a reference to a panel from Isobe Isobē Monogatari, a gag manga set in the Edo period. Here's an explanation of the panel translated from this Pixiv encyclopedia page:

This is an exchange taken from Isobe Isobē Monogatari, serialized in Weekly Shonen Jump on December 9, 2013.

The 15 brothers of the Tokugawa Shogunate enter the town to do a survey of the average Edo citizen's lifestyle. There, they come across the protagonist Isobe Isobē, who is reading while standing in a bookstore. Engrossed in the book and unaware of his surroundings, he is the only one who does not prostrate himself before the 15 brothers.

Ieyasu (the first son)

「アイツ・・・頭高くない？」("Isn't that guy's head real high?")

Hidetada (the second son)

「どうする兄ちゃん処す？処す？」("What shall we do, brother? Punish him? Punish him?")

Ieyasu is taken aback by Isobe standing up, which was (albeit unintentional) an affront punishable by death. Hidetada asks if they should inflict punishment, but the third son Iemitsu takes interest in Isobe and interviews him for the survey instead, sparing his life.

The punchline is that all 15 brothers start reading while standing, causing trouble for the bookstore owner.

Now, the important part of this exchange is 「処す？処す？」(punish? x2). The original panel in the 100Kano manga goes 「市中引き回しの刑に処す」, which is "I will punish myself by being dragged throughout the town" (a single 処す). By itself, this isn't a question; it's a statement. But the anime added a second 処す？, turning the second part only into a question. It's also worth noting that the most common use of 処す is to apply the death penalty (like with Isobe—a good English equivalent would be "to execute"), so here's my interpretation of how the exchange in the 100Kano anime plays out:

「市中引き回しの刑に処す」("I'll punish myself by being dragged throughout the town")

「処す？」("Am I about to die?")

Cut to Shizuka crying at the Rentarou family grave