Break sausage into an ovenproof skillet over medium heat; cook and stir, breaking the meat into small pieces as it cooks, until completely browned, about 10 minutes. Drain excess fat from the skillet.
Stir spaghetti sauce, cannellini beans, and thyme through the drained sausage. Bring the mixture to a simmer; cook at a simmer until hot, about 5 minutes. Remove skillet from heat; stir cheese through the mixture until melted evenly throughout.
Arrange crescent roll dough triangles atop the sausage mixture in a spiral pattern with points of dough toward the center, covering the sausage mixture completely.
Bake in preheated oven until crust is golden brown and meat mixture is hot and bubbly, about 12 minutes.