Bring 8 quarts water to a rolling boil and add 2 tablespoons salt.
Add the broccoli and cook until tender, about 6 minutes. Drain, reserve the cooking liquid and set aside.
In a 12-inch saucepan, heat the olive oil until hot but not smoking. Add the garlic and chile pepper halves and cook until soft and light golden brown, about 5 minutes.
While the garlic and pepper cook, cut the drained broccoli into small florets, about 1/2-inch each. Add the bread to the pan with a few ladles of reserved broccoli water. Then add the broccoli, season with salt and simmer 4 to 5 minutes. Garnish with extra-virgin olive oil and grated Parmesan. Serve very hot.