Three Cheese Lemon Spaghetti

Name: Three-Cheese Lemon Spaghetti
Ingredients: 1 lb. spaghetti; 2 tbsp. unsalted butter; 3 lemons, thinly sliced and seeds removed; 3 garlic cloves, grated; 2 tbsp. flour; 1.5 c. grated Parmesan; 1.5 c. grated Gruyere; 1.5 c. grated pecorino Romano; 2 egg yolks, room temperature; 0.25 c. chopped fresh parsley, plus more for garnish