Hot Italian turkey or pork sausage also works great here.
Preheat oven to 450 degrees with racks in upper and lower thirds. Place mushrooms stem-side up on a rimmed baking sheet; brush all over with 3 tablespoons oil. Season with salt and pepper. On another rimmed baking sheet, toss 4 cups bread with 2 tablespoons oil; season with salt and pepper. Roast mushrooms on top rack until softened, and bread cubes on bottom until golden, 12 minutes.
Combine sausage, egg, and remaining 2 cups bread; season with salt and pepper. Divide evenly among mushroom caps; roast until cooked through, 15 minutes more.
Adjust oven to broil. Divide cheese among mushrooms; broil until golden and bubbly, 3 minutes. Toss croutons with arugula, fennel, vinegar, and remaining 2 tablespoons oil; season with salt and pepper. Serve mushrooms, with salad alongside.