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Bring a pot of salted water to a boil. Cook pasta until al dente, about 11 minutes. Drain, reserving 1/2 cup cooking liquid.
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Meanwhile, melt 2 tablespoons butter in a medium skillet over medium heat. Cook garlic until fragrant, about 1 minute.
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Stir in breadcrumbs. Season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook, stirring, until breadcrumbs are golden brown, 4 to 5 minutes.
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Melt remaining 3 tablespoons butter in a large high-sided skillet over medium-high heat.
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Add cabbage, 1/2 teaspoon salt, and1/4 teaspoon pepper. Toss to coat.
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Cook, stirring, until slightly wilted, 3 to 4 minutes.
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Add 1/2 cup water.
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Cook, covered, until tender, about 4 minutes. Uncover, and let any water in the skillet evaporate.
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Stir in cream. Cook until sauce is reduced and thick enough to coat cabbage, about 1 minute.
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Season with salt and pepper.
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Add pasta and reserved 1/2 cup cooking liquid to cabbage.
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Cook for 1 minute.
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Stir in cheese.
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Transfer to a platter. Sprinkle breadcrumb mixture over top, and garnish with more cheese.