Preheat oven to 350 degrees F (175 degrees C).

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Heat 1 teaspoon olive oil in a skillet over medium heat; cook and stir mushrooms, shallot, thyme, garlic powder, sage, black pepper, and salt in the hot oil until liquid has evaporated and mushrooms and shallots are softened, 5 to 10 minutes. Add parsley; cook and stir for 1 minute. Mix in spinach; cook and stir until spinach is wilted, about 5 minutes. Stir in mustard. Remove from heat.

Place pork tenderloin on a work surface; lay prosciutto atop tenderloin. Spread mushroom-spinach mixture over prosciutto, leaving 1/2-inch border on all sides. Tightly roll tenderloin around the filling and tie together with kitchen string to keep closed.

Heat 2 tablespoons olive oil in a large skillet over medium heat; place rolled tenderloin in the hot oil. Sear until all sides are golden brown, about 10 minutes. Transfer seared tenderloin to a 9×13-inch casserole dish.

Bake in the preheated oven until pork is no longer pink in the center, 25 to 30 minutes. An instant-read thermometer inserted into the center should read at least 160 degrees F (71 degrees C).