Turn the oven on and set the heat to 350 degrees. Use 1 teaspoon of oil to grease the baking pan.
Put the skillet on the stove, turn the heat to medium and when the skillet is hot, add the remaining 2 teaspoons of oil.
Add the onion and cook, stirring occasionally, until golden and softened, about 7 minutes. Turn the heat down to low, add the vegetables and potatoes, and cook until the vegetables are the tenderness you like, 5–10 minutes. Set aside for 10 minutes, or until cool.
To crack the eggs: hit the midpoint between the more-rounded (wide) end and the pointier end of the egg against a clean counter.
Hold the egg over the bowl with the cracked part facing up. Pry the shell open gently and allow the egg inside to slide into the bowl. Be sure to wash your hands with soap and water after handling raw eggs!
Beat the eggs gently with the fork until the mixture is pale yellow.
Add the onion mixture, basil or parsley, cheese, salt, and pepper and mix well.
Pour the mixture into the baking pan or pie plate and, once the oven temperature has reached 350 degrees, carefully move it to the oven.
Bake until the top is golden and the eggs are set, 25–30 minutes.
Set aside to cool and serve warm or at room temperature, or cover and refrigerate up to 2 days.