Preheat oven to 450°F with oven rack positioned 8 inches from heat source. Toss together mushrooms, thyme, 2 tablespoons of the oil, and 1/4 teaspoon each of the salt and pepper on a rimmed baking sheet. Sprinkle pork evenly with paprika and 1/2 teaspoon each of the salt and pepper. Place pork in center of baking sheet; spread mushroom mixture evenly around pork. Roast in preheated oven until a thermometer inserted in thickest portion of pork registers 130°F, 7 to 8 minutes.
Remove from oven; add Broccolini and shallots, and drizzle with remaining 1 tablespoon oil and 1/4 teaspoon each salt and pepper. Return to oven; roast at 450°F until pork reaches 145°F and Broccolini is almost tender, about 10 minutes.
Remove from oven; transfer pork to a cutting board, and let rest 5 minutes. Meanwhile, return baking sheet with Broccolini to oven, and increase oven temperature to broil. Broil until Broccolini is lightly charred and tender, about 5 minutes. Slice pork. Sprinkle pork and vegetables evenly with tarragon, and serve.