Preheat oven to 375˚F.
For quick Bolognese sauce: Pour oil into a skillet and place over medium-high heat. Add beef and sauté for 3 to 4 minutes. Season with salt and pepper. Add ham and continue to sauté for an additional 2 to 3 minutes. Remove beef and ham with a slotted spoon and drain all but 1 tablespoon fat from the skillet.
Place skillet back over heat and add onion and garlic and sauté for 2 to 3 minutes. Add carrot and celery and continue to sauté for 4 to 5 minutes.
Add oregano and basil and sauté for 1 minute. Season with salt and pepper.
Add meat mixture back into the skillet, along with the tomato sauce. Lower heat to simmer and cook together for about 15 minutes. Season with salt and pepper. Set aside.
To assemble: Spray a muffin tin with nonstick spray. Place a pasta sheet square into the bottom of each muffin well and top with a spoonful of Bolognese sauce. Top each well with a sprinkle of mozzarella and a small sprinkle of Parmesan. Repeat the process 2 more times, adding only half the amount of Bolognese on the top layer.
Place muffin tin into the oven and bake for 20 to 22 minutes, or until the cups fully bake through.
Score the perimeter of each cup with a pairing knife and allow cups to cool for about 10 minutes before popping out of the wells and serving with more Parmesan and fresh basil.