Preheat the oven to 350°F. Remove mushroom stems; finely chop and set aside. Arrange mushroom caps on a rimmed baking sheet.
In a medium bowl, combine the cream cheese, cheddar cheese, salt, and pepper. Set aside.
Set a medium skillet over medium high heat and add the oil. When the oiil is hot, add the chopped mushroom stems and jalapeno. Cook for 3 minutes, frequently stirring until mushrooms have released their liquid. Add the garlic and continue cooking for an additional 30 seconds. Remove from heat.
Stir the mushroom and cheddar mixture into the cheese mixture. Adjust seasoning to taste. Spoon heaping portions of filling into the mushroom caps.
In a small bowl, combine the Panko breadcrumbs, melted, butter, and paprika. Sprinkle over the filled mushrooms.
Bake for 20 minutes, or until lightly golden. Serve warm or at room temperature.