Buzz the Cheddar, Gruyere, Robiola, prosecco, parsley, mustard and garlic in a food processor until combined but still chunky and spreadable. Spread the mixture among the bread slices. Sandwich 2 slices, spread 1 tablespoon of the butter on the outside of the sandwich and season with salt and pepper. Repeat with the remaining ingredients.
Griddle the sandwiches over medium heat until golden brown on the first side. Flip over and continue to cook until the cheese is melted and the exteriors are crispy, about 5 minutes.