Preheat the broiler. Toast half of the bread slices at a time on a baking sheet. Rub both sides of the toast with the cut garlic and drizzle 1/2 cup EVOO over the slices.
While the bread is working, in a large skillet, heat the remaining 3 tablespoons EVOO, 3 turns of the pan, over medium-high heat. Add the onion and bell pepper and cook until slightly softened, 2 to 3 minutes. Add the broccoli florets and chicken broth and bring to a simmer. Cover the pan and cook until the broccoli is tender, about 8 minutes. Uncover and boil until the liquid evaporates, 1 to 2 minutes. Season to taste with salt and pepper.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1 minute. Whisk in the milk, bring to a boil, then lower the heat and simmer, stirring often, until thick and smooth, about 5 minutes. Season the sauce with salt, pepper and the nutmeg.
Preheat the oven to 450 degrees . Layer half of the bread in a 9-by-13-inch casserole; spread with half of the broccoli mixture and top with the ricotta, a few ladles of the sauce and the Parmigiano-Reggiano. Layer the remaining bread slices, broccoli mixture, remaining sauce and all of the provolone cheese on top. Bake until brown and bubbling, 10 to 15 minutes.