Grate the Cheddar in large shreds on a box grater, as you would grate carrots. For a large amount, you can use the carrot grater blade of your food processor. If you don’t have a panini press, you can make these in a sauté pan with a small skillet on top.
Lay 6 slices of the bread on a cutting board and brush each slice generously with the butter. Turn the slices over and spread each one with 1 1/2 tablespoons of the chutney. Place 1/2 cup of the cheddar evenly on each slice and place the remaining 6 slices of bread on top of each sandwich. Brush the tops generously with melted butter.
Heat a panini press (see Author Notes) and grill the sandwiches according to the directions for the machine, until the bread is nicely browned on both sides and the cheddar is starting to melt. Place the sandwiches on a cutting board and cut each one in half diagonally. Serve hot.