Turn the oven on and set the heat to 350 degrees. Lightly coat the pan with 1 teaspoon oil.
Put the skillet on the stove, turn the heat to medium, and carefully add the remaining 2 teaspoons 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.
Put the eggs, salt, and pepper in the bowl and, using the whisk or fork, mix well.
Add the onion mixture, basil, and cheese, and mix well.
Pour the mixture into the pan or skillet and carefully move it to the oven once the temperature has reached 350 degrees.
Bake until the top is golden and the eggs are set, 25-30 minutes. (To see if the eggs are set, use pot holders to jiggle the pan back and forth: you should not see the eggs moving around in a liquidy way.)
Serve warm or at room temperature, or cover and refrigerate up to 2 days.