Preheat the oven to 350 degrees F. Make the filling: Heat 2 tablespoons butter in a large skillet over medium-high heat. Add the onion and cook, stirring occasionally, until soft and lightly golden, about 5 minutes. Add the garlic, thyme and cayenne and cook 1 more minute. Add the chopped tomatoes, brown sugar and 1 1/4 teaspoons salt. Bring to a simmer and cook until the tomatoes just begin to soften, 4 to 5 minutes. Remove from the heat, then gently stir in the cherry tomatoes and flour. Transfer to a 2-quart baking dish and dot with the remaining 1 tablespoon butter.
Make the topping: Whisk the flour, baking powder, granulated sugar, 3/4 teaspoon salt, and black pepper to taste in a medium bowl. Add the butter and use a pastry cutter or your fingers to rub the butter into the flour until it resembles coarse meal with pea-size pieces of butter. Add the milk, mustard and thyme and gently mix with a fork just until a sticky dough forms, being careful not to overwork the dough.
Drop balls of dough over the tomato filling and brush the dough with milk. Place the cobbler on a baking sheet and bake until golden and bubbling, 50 minutes to 1 hour. Let rest 15 minutes before serving.