Make tomato compote: Heat oil in heavy medium skillet over medium heat. Add tomatoes and sugar. Stir until tomatoes are very tender and mixture is thick, about 15 minutes. Season tomato compote with salt and pepper.
Make fish: Melt 2 tablespoons butter in heavy large skillet over medium-high heat. Add mushrooms and chopped shallots. Sauté until liquid evaporates and mushrooms begin to brown, about 8 minutes. Cool completely.
Using electric mixer, beat 5 tablespoons butter in large bowl until light. Beat in egg, then parsley, 1 tablespoon thyme, 1 teaspoon salt and 1/2 teaspoon pepper. Mix in breadcrumbs and mushroom mixture.
Preheat oven to 450°F. Sprinkle sliced garlic, sliced shallots and remaining 1 teaspoon thyme over bottom of 15 x 10 x 2-inch glass baking dish. Add fish stock, then fish fillets. Sprinkle fish with salt and pepper. Spoon breadcrumb mixture atop fish, covering tops of fish completely and pressing slightly to adhere. (Can be prepared 8 hours ahead. Cover tomato compote and fish separately and refrigerate. Rewarm compote over medium-low heat before serving.)
Bake fish until opaque in center, about 15 minutes. Transfer to broiler. Broil just until topping is crisp and golden, about 1 minute.
Using slotted spatula, transfer fish to plates. Transfer cooking liquid to another medium skillet. Bring cooking liquid to simmer over medium heat. Add remaining 1 tablespoon butter and whisk just until melted and well blended. Season sauce to taste with salt and pepper. Spoon sauce around fish. Spoon dollop of tomato compote alongside and serve fish immediately.