Preheat oven to 400 degrees . Bring large pot of water to a boil, salt it, add the broccoli rabe and cook for 8 minutes; drain.
In a large skillet, heat about 1/4 cup EVOO, 4 turns of the pan, over medium heat. Add the chile and garlic and swirl for 1-2 minutes. Add the broccoli rabe and toss for 2-3 minutes. Turn off the heat.
In a medium bowl, combine the ricotta, pecorino, parmigiano, thyme and remaining 2 tbsp EVOO; season with salt and pepper. Place the bread on a baking sheet and slather both halves evenly with ricotta mixture. Top with the broccoli rabe and mozzarella (or provolone). Bake until melted and bubbly, about 10 minutes. Cut into large chunks to serve.