Bring a large pot of salted water to a boil. Add the rice, reduce the heat and simmer, stirring once or twice, until the rice is tender, 12 to 15 minutes. Drain, shaking the colander to remove any excess water. Spread the rice on a rimmed baking sheet and let cool.
Preheat the oven to 450 degrees F and place a baking sheet on the middle rack. Melt 2 tablespoons butter in a large skillet over medium-high heat. Add the onion and cook until soft, about 8 minutes. Add the sausage and cook, breaking it up with a wooden spoon, until browned, about 6 minutes. Tear the basil and add to the skillet along with the garlic and tomato paste. Increase the heat to high and cook, stirring, until the tomato paste browns, about 4 minutes. Stir in the chicken broth, scraping up any browned bits from the bottom of the skillet. Cook, stirring occasionally, until thickened, 15 to 20 minutes.
Meanwhile, grease the bottom and sides of an 8-inch springform pan with the remaining 1 tablespoon butter. Coat the pan with the breadcrumbs, tapping to remove any excess. Put the eggs and all but 3 tablespoons of the pecorino cheese in a small bowl and beat with a fork. Put the cooled rice in a bowl; add the egg mixture and stir to combine.
Transfer about two-thirds of the rice mixture to the prepared springform pan. Using moist fingers, pat the rice onto the bottom and up the sides of the pan, forming a 1/2-inch-thick layer. Place the provolone slices over the rice in the pan. Spoon about three-quarters of the sausage filling over the provolone, filling it to 1/2 inch from the top. Pat the remaining rice mixture on top to enclose the filling, then sprinkle with the remaining 3 tablespoons pecorino cheese. Put the pan on the hot baking sheet and bake until golden, about 20 minutes. Transfer to a rack and let cool, 5 minutes.
Run a small knife around the edge of the pan to loosen the timbale, then remove the side of the pan. Slide a spatula under the timbale and transfer it to a platter. Thin the remaining sausage filling with a splash of water and reheat. Serve with the timbale.