Slice the pork on an angle into 1-inch-thick pieces; season with 1/2 teaspoon salt.
Heat a large ovenproof skillet over medium-high heat; add 1 tablespoon olive oil. Add the onion, peppers and mushrooms and season with remaining salt; cook until the vegetables are crisp-tender and slightly browned, about 6 minutes. Transfer to a plate.
Add the remaining 2 tablespoons oil to the skillet. Add the pork and sear over high heat until browned, 2 to 4 minutes per side. Transfer the pork to the plate with the onion and peppers.
Reduce the heat to medium and add the garlic, sage and tomato paste to the skillet. Cook, stirring, until the tomato paste turns brick-red, about 1 minute.
Pour in the wine and bring to a boil. Add the broth and return to a simmer. Arrange the pork in a single layer in the skillet; add onion and peppers and sprinkle with the cheese.
Transfer to the oven and broil until the pork is cooked through, 4 to 7 minutes.