Preheat the broiler. Brush a 9-by-13-inch flameproof baking dish with some of the melted butter.
Lay 6 of the bread slices out on a baking sheet in 1 even layer. Brush the tops generously with butter. Broil, rotating the baking sheet as needed, until the bread is golden brown and toasted, 2 to 4 minutes. Flip the slices (untoasted side facing up). Put 2 slices of the Cheddar each on 3 pieces of the bread and 1 slice of the Swiss cheese each on the 3 remaining slices. Return the baking sheet to the broiler until the cheese completely melted, 1 to 2 minutes. Sandwich together each Swiss-topped bread slice with a Cheddar-topped slice. Repeat with the remaining bread, Cheddar and Swiss and more butter.
Trim the crusts off the sandwiches, then arrange the sandwiches in 1 layer in the prepared baking dish. Scatter the Parmesan over the top.
Scatter the tomatoes on the baking sheet, and toss with 1/2 teaspoon salt. Broil until they are very soft and the skins are wrinkled and browned in spots, 4 to 5 minutes. Scatter the tomatoes over the sandwiches.
Switch the broiler to bake, and preheat to 350 degrees F.
Whisk together the half-and-half, mayonnaise, eggs, 1 teaspoon salt and a few grinds of black pepper in a large bowl. Pour over the tomatoes and sandwiches (the liquid won't completely cover). Cover with foil, and bake until the custard is mostly set, about 30 minutes. Remove the foil, and continue to bake until the casserole is hot and golden brown in spots, about 15 minutes more. Scatter the parsley on top, and serve hot.
Save the trimmings from the grilled cheeses, and dip them in ketchup for a snack!
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