Preheat oven to 375°F (190°C).
Place the loaf on a cutting board. Using a sharp serrated knife, make parallel cuts about 1 inch apart from the top toward the bottom without cutting through the bottom crust. Turn the loaf 90° and make another set of parallel 1-inch-apart cuts to form a crosshatch, again not cutting through the bottom.
In a small bowl, mix the melted 4 tablespoons butter with the 3 tablespoons pesto until combined.
Gently open the cracks in the bread and drizzle the butter-pesto mixture into the cracks, using a spoon or pastry brush to work the mixture in between slices.
Use all of the 3/4 to 1 cup shredded mozzarella, stuffing pieces into the cracks and distributing the cheese evenly around the loaf. Use your fingers or a small spoon to push cheese into deeper cuts as needed.
Place the stuffed loaf on a sheet of aluminum foil. Fold up the foil sides and cover the loaf completely (use a second sheet of foil on top if needed). Set the wrapped loaf on a baking sheet.
Bake covered for 20–25 minutes, until the cheese in the center is melted. Remove the top foil and bake an additional 5 minutes to lightly brown the top.
Carefully remove from the oven and unwrap (steam will be hot). Sprinkle with freshly chopped parsley to garnish. Let sit 2–3 minutes, then slice through the cuts and serve.