Pre-heat the oven to 425?F.
Combine the EVOO and garlic in a small saucepot. Cook the garlic until soft and fragrant, about 5 minutes. Meanwhile, par-roast the mushroom caps brushed with garlic oil and seasoned with salt and pepper, cap-side down, until just tender, 10 minutes. Cool.
Place the leftover stuffing in a mixing bowl and mix with the beaten egg. Stuff the mushroom caps and top each mushroom with a mix of white sharp cheddar, Parmigiano Reggiano and finely chopped fresh sage. Bake for 20 minutes.