On a cutting board, use a sharp knife to slice the goose breasts thin (about ¼-inch-thick) against the grain.
Transfer sliced meat to a bowl, then season with salt, garlic powder, oregano, and pepper. In a large skillet over medium-high heat, add 2 teaspoons of the avocado oil and cook the goose until fully cooked, about 3 minutes. Remove from the skillet and transfer to a large plate.
Sauté mushrooms, peppers, and onion in the same skillet with 1 teaspoon of oil until tender, about 5 minutes. Transfer to the same large plate with the cooked goose meat and toss to mix.
Spread 1 teaspoon of mayonnaise on each tortilla. Place ½ cup of the meat and vegetable filling onto one side of the tortilla. Top with 2 slices of cheese and close the tortilla.
In the same skillet, over medium heat, add 1 teaspoon of oil. Cook two quesadillas at a time for about 2 minutes per side, or until crispy and the cheese is melted.
Remove from skillet, cut each quesadilla into thirds. Repeat with remaining quesadillas.