In a saucepan, melt together the olive oil, butter, garlic, chili flakes, and paprika. Let the butter bubble up and begin to brown, then remove from heat.
Place the eggs in a medium pot and cover with water by 1 inch. Bring to a boil over high heat, cover, and remove from the heat. Let sit 5 minutes for a softer yolk and 10 for hard boiled.
Drain the eggs, peel and slice into quarters. Add to a bowl with the cheese and season with salt. Use a fork to lightly mash the eggs and cheese together.
Spread the avocado on toast. Spoon over the eggs, then drizzle with chili butter.
