Preheat oven to 375 degrees F. In a large skillet, heat oil over medium-high. Cook sausage, breaking it up with a fork, until no longer pink, about 5 minutes. Add garlic, shallot, and chopped mushrooms stems; season with salt and pepper. Reduce heat to medium, and cook until vegetables are tender, stirring occasionally, 4 to 6 minutes. Transfer to a bowl, stir in parsley and oregano, and let cool.
Add egg, breadcrumbs, and cheese to cooled sausage mixture, and stir to combine. Place mushroom caps on a rimmed baking sheet. Season cavities with salt and pepper, then stuff with sausage mixture, dividing evenly and packing tightly. Drizzle lightly with oil.
Bake until mushrooms are tender and tops are browned, 25 to 30 minutes. Serve immediately garnished with additional oregano leaves, if desired.