Crack the eggs into a medium bowl. Add the milk, salt, and pepper. Whisk vigorously until the mixture is homogenous and slightly frothy (about 30 seconds). Slice the tomato thinly (about 1/8 to 1/4 inch). If using, shred cheese and set pesto and basil leaves within reach.

Season tomatoes (optional)

Place the tomato slices on a small plate, drizzle with the 1/2 tsp olive oil and a light pinch of salt and pepper. This step brightens the tomatoes and is optional but recommended.

Heat pan

Warm a nonstick or well-seasoned 8-inch skillet over medium-low heat. Add the butter and let it melt, swirling to coat the pan evenly. The butter should foam but not brown.

Cook eggs

Pour the whisked eggs into the skillet. Let them sit undisturbed for about 15–20 seconds until the edges just begin to set. Using a silicone spatula, gently pull the cooked edges toward the center while tilting the pan to let uncooked egg flow to the edges. Repeat for about 45–60 seconds until the surface is mostly set but still slightly glossy on top.

Add fillings

Evenly sprinkle the shredded cheese (if using) over one half of the omelet. Arrange the tomato slices on top of the cheese (or directly on the egg if not using cheese). Spoon the 2 tablespoons of basil pesto in small dollops over the tomato slices so the pesto will melt into the warm omelet when folded.

Fold and finish

Carefully fold the empty half of the omelet over the filled half using the spatula. Let the folded omelet cook for another 30–45 seconds to warm the fillings and finish setting the interior without overcooking. If you prefer the tomatoes warmed through, cover the pan for 20–30 seconds on very low heat.

Plate and garnish

Slide the omelet onto a plate. Tear the fresh basil leaves and scatter over the top for garnish. Optionally add an extra small drizzle of pesto or a grind of black pepper. Serve immediately.