Beat eggs with salt, pepper. Set aside, and cook vegetables first.
Heat pan to medium high. Add butter and oil to hot pan.
Reduce heat, add brussels sprouts and potatoes, and cook until softened.
Add in arugula.
Add beaten eggs. Using a large spatula, allow the eggs to run under the frittata mixture. Keep lifting and moving the mixture. Once you have no more runny eggs, and the remains to be cooked, put in oven for about 5 to 10 minutes. You want to make sure you don't see any runny egg mixture.
Cut into wedges and eat hot or at room temperature.