Preheat oven to 400°F. Toss tomatoes with a drizzle of olive oil, garlic, salt, pepper and thyme; sprinkle with a drizzle of vinegar. Roast until tomatoes are slumped, about 30 minutes; remove.
In a deep-sided skillet, bring a few inches of water to a boil. Salt water and boil broccoli rabe for 4-5 minutes; cold shock and drain well. Chop about 1 (packed) cup greens and reserve the rest for another use (like a frittata).
Combine sausage, chopped rabe, Pecorino, ground sirloin, grated onion, sage, salt and pepper. Form 4 patties that are thinner at center and thicker at edges for even cooking.
Heat 2 tablespoons oil in a skillet over medium-high heat. Add burgers and cook about 10 minutes turning once; top with cheese in last minute or so and tent to melt.
Serve burgers on toasted buns topped with roasted tomatoes.