In a large bowl, toss the halved tomatoes with 2 Tbsp of olive oil, salt and pepper. Spread tomatoes on a rimmed baking sheet; bake at 400° for 20 minutes. Place tomatoes in a large bowl; set aside.
Cook pasta according to package directions. Drain cooked pasta, but reserve ¼ cup of pasta water after cooking.
Meanwhile, place sausage with the remaining 1 Tbsp of olive oil in a large skillet and cook over medium-high heat until fully cooked. Add garlic and saute for 1-2 more minutes, stirring often.
Drain excess fat and transfer sausage and garlic into bowl with tomatoes.
Add oregano and cooked pasta to the bowl and toss until well-combined. If needed, add a bit of the reserved pasta water to loosen mixture.
Place pasta onto plates and top with burrata and basil. Finish with freshly ground black pepper and a light drizzle of olive oil before serving.