Position a rack in the center of the oven and heat the oven to 425°F.
Trim and slice each tenderloin on the diagonal into 3 thick medallions. Place each medallion on a cut side, and using your hands, gently press on each to flatten slightly. Season on both sides with salt and pepper.
Heat the butter and oil in a 12-inch ovenproof skillet over medium-high heat. Working in batches, cook the pork, flipping once, until golden-brown, 2 to 3 minutes per side. Transfer to a plate and set aside.
Turn the heat down to medium and add the onion and 1/4 tsp. salt. Cook, stirring often, until lightly browned, 5 to 6 minutes. Add the cabbage, 1/4 tsp. salt, and 1/2 cup water. Stir, cover, turn the heat down to medium low, and cook, stirring occasionally, until the cabbage is tender, 15 to 20 minutes. Stir in the chickpeas and the reserved cooking liquid and season to taste with salt and pepper. Arrange the pork over the cabbage and chickpeas and top with the prosciutto and fontina. Transfer the skillet to the oven and cook until the pork reaches 145°F, 12 to 15 minutes. Let rest for 5 minutes before serving.