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Melt butter in a large skillet over medium heat, and cook and stir onions until translucent but not browned, about 8 minutes. Mix cooked onions and butter in the skillet with rice; stir in half-and-half, salt, and 1 3/4 cup of Swiss cheese. Transfer mixture to the prepared baking dish, and sprinkle remaining 1/4 cup Swiss cheese over top.
Bake uncovered in the preheated oven until rice is tender and top is lightly browned, about 1 hour.