Spray a 9 x 9-inch pan with nonstick spray. Set aside.
Bring broth to a boil in a large pot. Slowly whisk cornmeal into broth.
Reduce heat to a low simmer, then cook (stirring frequently) until polenta begins to pull away from the bottom of the pot when you stir it (about 15 minutes).
Remove polenta from heat, then stir in cheese, butter, salt, and pepper.
Pour polenta into the 9 x 9-inch pan, smooth the top, then refrigerate until chilled (about 1 hour).
Prepare sauce:
Heat olive oil in a large skillet over medium-high heat.
Add peppers to oil and cook, stirring frequently, until lightly browned.
Stir mushrooms into peppers and cook until browned, stirring often. Season with salt and pepper.
Reduce heat to medium and stir in marinara sauce.
Heat sauce until warmed, then add spinach and basil, and cook until wilted. Remove from heat.
Assemble dish:
Preheat oven to 425.
Spray a 9 x 13-inch (or similar size casserole dish) with nonstick spray.
Turn polenta out of pan onto a cutting board, then cut into 6 pieces.
Place polenta in pan, overlapping if necessary. Top with sauce and then sprinkle with mozzarella.
Bake until hot and cheese is browned in spots, about 15 to 20 minutes.