For the pork and prosciutto: Cut the pork into 1-inch-thick medallions. Place each piece between 2 pieces of plastic wrap and using the flat side of a meat mallet, pound to 1/8-inch even thickness.
Add the flour to a shallow dish and season liberally with salt and pepper. Lightly dredge each piece of pork in the flour mixture and place on a wire rack to set.
Add the olive oil to a 12-inch cast-iron skillet and set over medium heat. Working in batches, add 3 to 4 pieces of prosciutto and cook on each side until golden and crispy, 30 to 45 seconds. It’s okay if they curl up a little and twist as they cook; the most important thing is they are crispy! Set aside on paper towels to drain. Repeat with the remaining prosciutto. Add the sage leaves to the oil and cook until translucent and crispy, about 45 seconds. Drain and set aside.
Cook the pork in the same skillet in batches, 3 to 4 pieces at a time. Cook until golden brown, 1 to 2 minutes per side. Set out on a serving platter and keep warm.
For the marsala brown butter: Discard all but 2 tablespoons of oil from the skillet. Remove any black bits with a paper towel. Add the butter, garlic and shallots and cook over medium heat until the butter browns and the shallots turn translucent, 3 to 4 minutes. Use a wooden spoon to scrape up any brown bits from the bottom of the skillet.
Move the skillet away from the heat and deglaze with the Marsala. Return to the heat and simmer until reduced by half, 1 to 2 minutes. Add the chicken stock and simmer for another 2 minutes. Season with salt and pepper, then add the chopped parsley.
To serve, pour the brown butter sauce over the pork and drizzle with lemon juice. Garnish with the crispy prosciutto and sage.