In a large pot of salted boiling water, cook pasta until just cooked through, 7 to 9 minutes.
Meanwhile, trim brussels sprouts and slice very thin. Heat oil in a large nonstick pan over medium-high heat. Add sprouts and 1 tsp. salt and cook, stirring occasionally, until softened and beginning to brown, 8 to 10 minutes. Add garlic, 2 tbsp. butter, and 1/4 tsp. pepper and cook 30 seconds more. Transfer sprouts to a bowl. Wipe pan clean with a paper towel.
Return pan to medium-high heat and add remaining 1 tbsp. butter. When it's melted, crack in eggs and cook, covered, until whites are just set and butter is starting to brown, about 2 minutes. Season eggs with remaining 1/4 tsp. each salt and pepper. Cover to keep warm.
Reserve 1 cup pasta-cooking water. Drain pasta and return to pot. Add sprouts, three-fourths of reserved cooking water, and lemon zest and juice. Cook, stirring, until everything is heated through. Add more cooking water if necessary. Add half of cheese and stir to just combine.
Divide pasta among 4 wide, shallow bowls and top each with some of the remaining cheese and an egg.