Heat the oil in a skillet over medium heat. When the oil is hot, add the onion and cook, sprinkling with salt and pepper to taste, for 3 minutes, or until it’s soft. Add the vegetables, raise the heat, and cook, stirring occasionally, until they soften, anywhere from a couple of minutes for greens to 15 minutes for sliced potatoes. Adjust the heat so the vegetables brown a little without scorching. (If you’re starting with precooked vegetables, add them to the onions and give a couple of good stirs before proceeding to the next step.)
When the vegetables are nearly done, reduce the heat to low and add the basil, if using. Cook, stirring occasionally, until the pan is almost dry, up to another 5 minutes for wetter ingredients like tomatoes or mushrooms.
Meanwhile, beat the eggs with salt and pepper to taste, along with the cheese. Pour over the vegetables, using a spoon if necessary to distribute them evenly. Cook, undisturbed, for 10 minutes, or until the eggs are barely set. (You can set them further by putting the pan in a 350°F oven for a few minutes or by running it under the broiler for a minute or two.) Cut into wedges and serve hot, warm, or at room temperature.