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Preheat oven to 350 degrees. Butter a 2-quart shallow baking dish; set aside.
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Place carrots in a large pot and add enough water to cover by 1 inch. Generously salt water and bring to a boil over high heat. Reduce heat and simmer until carrots are tender and easily pierced with the tip of a sharp knife, about 10 minutes.
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Strain carrots and transfer to the bowl of a food processor; process until pureed. Transfer carrot puree to a large bowl; stir in milk, cracker crumbs, cheese, onion, butter, salt, black pepper, and cayenne pepper.
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In the bowl of an electric mixer fitted with the whisk attachment, whisk eggs until foamy. Whisk eggs into carrot mixture until just combined.
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Transfer carrot mixture to prepared baking dish and bake until puffed and light golden brown on top, 40 to 45 minutes; serve warm.