Place one potato on a cutting board then place a wooden spoon on either side as a guide. Using a sharp knife, cut thin 1/8 inch slices down the potato, stopping at the wooden spoon. Leave about 1/2 an inch on ends unsliced. Place sliced potatoes in a baking dish or cast iron skillet.
Brush potatoes with melted butter. Drizzle with olive oil then sprinkle with salt and pepper.
Bake uncovered for 40 minutes.
Make the pesto by adding basil and walnuts to a food processor. Pulse until combined then add garlic, Parmesan, salt, and lemon juice. Pulse until combined well then drizzle in olive oil while processor is running. Continue processing until mixture is smooth. Add more olive oil if needed to reach desired consistency. Taste and add more salt if needed. Pour into a container and set aside.
After 40 minutes of baking, remove potatoes from oven and spoon a couple tablespoons of pesto over each one. Top each potato with mozzarella. Place back in oven and bake for an additional 5-10 minutes or until cheese is melted and bubbly.