Start by spreading about 1 tablespoon of pesto on the inside surface of each bread slice. Use the back of a spoon to spread it evenly. The goal is a thin, flavorful layer — not too thick or the bread will become greasy.
On two of the pesto-coated slices, arrange 2–3 slices of fresh mozzarella, enough to cover but not overflow. Then place the remaining two slices of bread on top, pesto side down. Press gently so the layers hold together before transferring them to the skillet.
Melt 1½ tablespoons of butter in a skillet over medium-low heat. Place the sandwiches in the skillet, cover with a lid, and cook for about 3–4 minutes per side until golden brown. Add the remaining butter when flipping. The lid helps melt the cheese evenly while keeping the bread perfectly crisp. Once both sides are golden, remove from the heat and cut diagonally before serving.