Preheat oven to 425 degrees F (220 degrees C). Arrange potato nuggets in a single layer on a large baking sheet.
Bake in the preheated oven, tossing often, until golden brown and crisp, about 35 minutes. Remove from oven; sprinkle Mexican cheese on top. Continue baking until cheese is melted, about 10 minutes more.
Drizzle sausage gravy over potato nuggets and cheese. Scatter avocado, andouille sausage, green onions, and jalapeno peppers on top.
Grease a large skillet with cooking spray. Crack in 3 eggs; cook until egg whites are firm and starting to brown on the bottom, about 5 minutes. Transfer to a large plate using a wide spatula. Repeat with remaining eggs. Arrange fried eggs over potato nuggets.