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1
Preheat the broiler.
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2
Brush a 9-by-13-inch baking dish with some of the melted butter.
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3
Lay 6 of the bread slices out on a baking sheet in 1 even layer.
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4
Brush the tops generously with butter.
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5
Broil, rotating the baking sheet as needed, until the bread is golden brown and toasted, 2 to 4 minutes.
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6
Flip the slices (untoasted side facing up).
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7
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.
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8
Return the baking sheet to the broiler until the cheese completely melted, 1 to 2 minutes.
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9
Sandwich together each Swiss-topped bread slice with a Cheddar-topped slice.
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10
Repeat with the remaining bread, Cheddar and Swiss and more butter.
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11
Trim the crusts off the sandwiches, then arrange the sandwiches in 1 layer in the prepared baking dish.
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12
Scatter the Parmesan over the top.
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13
Scatter the tomatoes on the baking sheet, and toss with 1/2 teaspoon salt.
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14
Broil until they are very soft and the skins are wrinkled and browned in spots, 4 to 5 minutes.
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15
Scatter the tomatoes over the sandwiches.
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16
Switch the broiler to bake, and preheat to 350 degrees F.
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17
Whisk together the half-and-half, mayonnaise, eggs, 1 teaspoon salt and a few grinds of black pepper in a large bowl.
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18
Pour over the tomatoes and sandwiches (the liquid won't completely cover).
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19
Cover with foil, and bake until the custard is mostly set, about 30 minutes.
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20
Remove the foil, and continue to bake until the casserole is hot and golden brown in spots, about 15 minutes more.
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21
Scatter the parsley on top, and serve hot.
