Creamy Bucatini with Seared Brussels Sprouts

Name: Creamy Bucatini with Seared Brussels Sprouts
Ingredients: 1 pound bucatini; 1/2 pound brussels sprouts, trimmed and halved; 2 tablespoons neutral oil (canola, vegetable, grapeseed, etc.); Sea salt; Freshly ground black pepper; Juice of one lemon; 5 tablespoons butter, divided; 1 clove garlic, minced; 1/2 cup stock; 3/4 cup finely grated pecorino; 1 egg