Preheat oven to 375°F. Cut tomatoes in half. Squeeze out the seeds and lay on a sheet pan lined with foil, cut side up. Drizzle with olive oil and season with the thyme and salt and pepper. Bake in the oven for about 45 minutes to an hour until tomatoes have begun to shrivel.
To make burgers, heat a grill to high heat. Break the ground lamb up into small pieces in a mixing bowl. Toss with salt and pepper and form into thin patties about 4 ½” to 5” across. Place on a hot grill. Lightly char on one side, turn to the other side and place the crumbled Gorgonzola cheese on top. Close the lid and cook to medium or desired doneness. Split the potato buns and toast on the grill. Place some arugula on the bottom of each bun, top with the tomatoes, lamb burgers and close with the top of the buns.