Heat 1 1/2 tablespoons of the butter in a large nonstick skillet over medium-high heat. Add onion; cook, stirring occasionally, until lightly browned, about 3 minutes. Reduce heat to medium; cook, stirring often, until onion is caramelized, about 10 minutes; add garlic in final 2 minutes. Add kale, salt, and pepper. Cook, stirring, until kale has wilted, about 3 minutes. Transfer to a plate.
Increase heat to medium-high. Add remaining 1 1/2 tablespoons butter to skillet; cook until lightly browned, about 1 minute. Add squash and nutmeg; spread in a thin, even layer. Cook, without stirring, until deeply browned on bottom, about 5 minutes. Gently stir; cook, without stirring, for 1 minute. Gently stir in onion mixture; remove from heat.
Heat oil in a large skillet over high heat. Break eggs into skillet; cook for 3 minutes for sunny-side up or to desired doneness.
Divide hash among 4 plates. Top with eggs.
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