Preheat the oven to 400°F (200°C).
Make the filling: in a mixing bowl combine ricotta, 1/3 cup Parmesan, 2 teaspoons basil pesto, a pinch of salt and pepper, and 2 tablespoons chopped fresh basil; stir until uniform.
If using dry lasagna sheets, ensure they are cooked and cooled; if using fresh sheets, proceed as-is.
Spread about 2 tablespoons of the ricotta filling evenly over each lasagna sheet using the back of a teaspoon, leaving a small border at the edges.
Roll each sheet tightly into a tube and set aside; you should have six roll-ups total.
Prepare the baking vessel: spoon about 3/4 cup of marinara (rosa) sauce into the bottom of each oven-safe mug or small deep dish, then sprinkle a layer of shredded mozzarella over the sauce.
Place three rolled lasagna sheets into each mug standing upright on the cheese layer, pressing down gently so they fit snugly.
Spoon about 1/2 cup additional marinara sauce over the top of each set of rolls, cover each mug with foil, and bake for 15–20 minutes until heated through and cheese is melted; total oven time is included in cook time.
Carefully remove the foil, invert each mug onto a plate by placing the plate over the mug and flipping swiftly, then lift the mug so the roll ups sit on the plate.
Sprinkle the top with the remaining 2 tablespoons freshly grated Parmesan and extra chopped basil, then serve immediately.