In a small bowl, whisk together mayonnaise, lemon juice, garlic, parsley, salt, pepper, and red pepper. Stir in 2 tablespoons of the shredded cheese.
Spread mayonnaise mixture onto sourdough, and place bread onto a parchment-lined baking sheet. Top with broccoli, and divide remaining cheese onto the two slices.
Broil for 3-5 minutes, until cheese is melted and bubbly.