In a small nonstick skillet, heat 1 tablespoon olive oil over medium heat. Stir in the tomato paste, paprika, and cayenne and fry until the oil is red and fragrant, about 2 minutes. Set aside to cool.
Separate the egg yolks from the whites. Save yolks for another use and transfer the whites to a small bowl. Add the seasoned oil (reserve the skillet), along with a pinch of salt and lots of black pepper. Using a fork or small whisk, beat the eggs and oil until combined.
In the reserved skillet (no need to wipe it out), heat ½ tablespoon olive oil over medium-low. Pour in the eggs. Let sit undisturbed until the edges start to bubble and cook, just a minute. Then run your spatula across the bottom of the pan to spread the eggs. Continue moving the eggs around until the surface is mostly set when you tilt the pan, about 4 minutes. Cover with a lid or foil and cook until the eggs are set on top, another minute.
Cut the frittata in half, then place on one piece of toast. Thinly slice the avocado, place half on top, and season with salt and pepper. Repeat with remaining frittata, avocado, and toast.