Place the butter in a small oven-safe pan (a 6-inch round cake pan works perfectly). Set the pan inside your air fryer basket and preheat at 300°F for 1 minute, until the butter is melted.
While the butter melts, crack the eggs into a bowl. Add the milk (if using), salt, and pepper. Whisk vigorously for at least 20 seconds, until the mixture is uniform and slightly frothy.
Open the basket, swirl the melted butter to coat the pan, and pour in the eggs. Close the basket and cook at 300°F for 2 minutes.
Open the basket and gently stir the eggs with a silicone spatula, pushing the set edges toward the center. Cook for 2 more minutes.
Stir again, breaking up any remaining set patches. Cook for a final 1 to 2 minutes, until the eggs are just set but still look slightly soft and glossy.
Remove from the air fryer immediately — the residual heat will finish cooking them. Serve right away.