Heat half of the oil in a 9 or 10-inch skillet on medium low heat. Add the onion and cook, stirring every so often, until the onions are soft and very translucent, about 10 minutes.
Meanwhile, remove and discard the stems from the spinach and chop it coarsely. When the onions are ready, stir in the spinach, adding one third at a time, and stirring the spinach until it wilts to make room for more spinach. When all of the spinach has wilted and practically disappeared, season generously with salt and pepper.
In a bowl, beat the eggs with a whisk. Add the garlic, the wilted spinach, the thyme and the cup of cheese.
Wipe the skillet with a paper towel to clean out any spinach/onion residue. Heat to medium flame, adding the other half of oil. Pour in the egg and spinach mixture. Set the timer for 10 minutes. You don’t have to stir the eggs, just let them cook until they set.
Preheat the broiler. After 10 minutes the eggs should be cooked with a bit of liquid on the top. Sprinkle the cheddar cheese over those moist top eggs and slide the skillet under the broiler, about 6 inches from the flame. Cook until the cheese is golden brown, about 3 minutes.
Cut into wedges to serve.