Heat oil in a large skillet over medium-high. Add sausage and cook, stirring occasionally, until lightly browned, about 2 minutes.
Add tomatoes, garlic, and thyme, stir to combine, and cook, pressing down on some of the tomatoes with the back of a wooden spoon to encourage bursting, 2 minutes.
Add broccolini; season with salt and pepper. Cook, covered, 2 minutes. Remove from heat and stir in butter until melted.
Divide sausage mixture among plates. Serve with bread alongside, if desired.