Position a rack in the top third of the oven and preheat to 325 degrees F. Grease a 2-quart oval gratin or baking dish with 1 tablespoon of the butter.
Whisk the eggs and half-and-half together in a large bowl. Add the sugar, corn, 1/2 cup of the crushed crackers, 1/2 cup of the Cheddar and 1 teaspoon salt. Pour the mixture into the baking dish, spreading with a spatula or spoon to make an even layer.
Bake uncovered until the edges are set but the center is still wet, about 35 minutes. Meanwhile, melt the remaining 3 tablespoons butter in a small dish in the microwave, about 30 seconds. Mix the butter into the remaining 1 cup crushed crackers until evenly moistened. Sprinkle the remaining 1/2 cup shredded Cheddar and the buttered crackers over the corn. Continue baking until the edges are brown and the top is set and golden, 20 to 25 minutes more. Remove from the oven and let stand 5 minutes before serving.
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