Stir together softened butter, minced garlic, parsley, Parmesan, garlic powder, oregano, salt, pepper, and red pepper flakes until creamy.
Spread a thin layer of mayonnaise on the outside of each bread slice.
Flip the slices and smear the garlic-butter mixture on the inside faces.
Pile mozzarella and cheddar on two slices. Top with remaining slices, mayo side facing out.
Warm a nonstick skillet over medium-low heat.
Set sandwiches in the skillet and cook until deep golden, 3–4 minutes per side. Cover briefly to melt cheese if needed.