Position an oven rack in the middle and preheat the oven to 350°F (175°C). Place two small oven-safe dishes (ramekins) in the oven to warm while it heats.
Crack two eggs into each of two small bowls and season lightly with salt and pepper; set aside.
Remove the hot dishes from the oven. In each dish add 1/2 tablespoon butter, 1 tablespoon half-and-half, and half of the minced garlic; whisk briefly to combine.
Turn the oven broiler on and return the dishes to the oven just until the butter mixture begins to bubble but does not burn, about 1–2 minutes.
Remove the dishes from the oven and gently slide the two eggs from each bowl into its prepared dish. Sprinkle 1–2 tablespoons Parmesan and 1–2 tablespoons chopped basil over each dish.
Place the dishes under the broiler and cook until the egg whites are set and yolks are still runny, about 2–4 minutes, or cook to your desired doneness.
Remove from the oven and, if desired, top with additional Parmesan and basil. Serve immediately.
