Preheat oven to 400°F. Prepare a baking sheet with a lining of parchment or a coating of nonstick cooking spray.
Gently wipe or rinse mushrooms to remove surface dirt. Pat dry with a paper towel. Remove the stems and, if desired, carefully scrape out some of the gills with a small spoon to make more room for the filling.
Toss mushroom caps in olive oil until coated, then place them hollow side up on the prepared baking sheet. Optionally season them lightly with salt and pepper.
Bake mushroom caps for 5 minutes to release excess moisture. Remove from oven and drain any liquid from the baking sheet before stuffing.
In a large bowl, mix together the ham, cheddar cheese, cream cheese, and green onions until well combined.
Divide the filling equally among the mushroom caps, mounding slightly.
Bake for about 15-20 minutes, or until the cheese is melted and bubbly and the mushrooms are tender.
Remove from oven and let stand for a 2-3 minutes to firm up the filling and make them easier to handle. Serve hot.