Place oven rack in bottom third of oven. Preheat oven to 400°F. In large skillet, heat 2 tbsp of the oil, honey and balsamic vinegar over medium heat, stirring to combine. Add tomatoes, half of the basil and 3 thyme sprigs. Cook over low heat for 2 minutes, turning tomatoes over halfway through cooking.
Line pie plate with parchment paper; arrange puff pastry, pressing into edges. Sprinkle bottom with breadcrumbs and Parmesan; top with tomatoes.
Bake until crust is browned and tomatoes are cooked through, about 30 minutes.
To serve, arrange Mozzarella, sliced or whole, remaining thyme and remaining basil on top. Drizzle with remaining oil; sprinkle with salt and pepper.