Preheat the broiler. Season the pork with salt. Heat 2 tablespoons olive oil in a large ovenproof skillet over medium-high heat. Working in batches, dredge the pork in the flour and add to the skillet; reserve the flour. Cook until browned, 1 minute per side. Transfer to a plate.
Reduce the heat to medium; add the shallot to the skillet and cook, stirring, until soft, 3 minutes. Increase the heat to medium high. Add the sage and 2 teaspoons of the reserved flour. Cook until sizzling, 30 seconds. Add the wine; bring to a boil. Cook until reduced by half, 2 minutes. Add 1 cup broth; return to a boil and cook until thickened, 2 minutes. Add the pork; return to a simmer. Top each chop with a prosciutto slice, then the cheeses. Broil until bubbling, 1 to 2 minutes.
Heat the remaining 1 tablespoon olive oil in a small saucepan over medium heat. Add the chopped prosciutto; cook until crisp, 1 to 2 minutes. Add the peas, the remaining 1/4 cup broth and 1/4 teaspoon salt. Cover; cook until the peas are tender, 5 minutes. Uncover; increase the heat to medium high. Cook until the liquid is reduced, 1 minute. Serve with the pork.