Heat olive oil in a skillet over medium-high heat.
Add mushrooms to the hot oil, sprinkle with salt, and cook and stir until liquid from mushrooms has evaporated.
Stir butter into mushrooms, reduce heat to medium, and cook and stir mushrooms until golden brown, about 5 minutes.
Stir onion into mushrooms and cook, stirring often, until onion is translucent, 5 minutes. Remove skillet from heat and stir garlic into mushroom mixture until fragrant, about 1 minute. Transfer mixture to a mixing bowl.
Mix oats into mushroom mixture until thoroughly combined. Gently stir 1 ounce Parmigiano-Reggiano cheese into mixture.
Add bread crumbs, 1/4 cup parsley, and 1 egg; season with salt, black pepper, cayenne pepper, and oregano.
Mix together with a fork until crumbly. Stir in remaining 1 egg. Mixture should hold together when pressed.
Cover bowl with plastic wrap and refrigerate at least 4 hours. For best flavor and texture, refrigerate overnight.
Preheat oven to 450 degrees F (230 degrees C). Line a baking sheet with a silicone baking mat or parchment paper.
Form mixture into small meatballs using a 2-tablespoon scoop. Roll meatballs lightly between your hands until smooth, if desired; arrange meatballs on prepared baking sheet.
Bake in the preheated oven until meatballs are lightly golden brown, 12 to 15 minutes.
Bring pasta sauce to a boil in a large saucepan; reduce heat to low. Gently stir meatballs into sauce until coated.
Simmer meatballs in sauce until cooked through, 45 minutes to 1 hour. Transfer to a serving bowl and garnish with 1 tablespoon Parmigiano-Reggiano cheese and 1 tablespoon parsley.