Melt the butter in a large nonstick skillet over medium-low heat, swirling to coat the bottom of the pan.
Crack the eggs directly into the skillet and season with salt and freshly ground black pepper.
Reduce the heat to low. As the whites begin to set, gently move the uncooked egg whites around the skillet with a spatula to help them cook evenly without breaking the yolks.
Once the whites are mostly set, arrange the tomato slices between the yolks and sprinkle the shredded cheddar evenly over the top.
Cover the skillet and cook over the lowest heat for 3–5 minutes, until the cheese is melted and the egg whites are fully set. Cook a little longer if you prefer firmer yolks.
Slide the eggs out of the skillet onto a large serving plate and serve hot immediately. Bon appétit!