To saute onions, heat cooking oil or butter over medium-high heat in a large pan or skillet until it becomes hot. Use 1 tablespoon of oil or butter per onion. Place the cut onions in the skillet, and cook five to seven minutes, or until the onions are tender. Stir the onions frequently while cooking using a wooden spoon or spatula that is heat-proof. Cook longer if desired until the edges of the onion slices are beginning to brown.
Take care not to crowd the onions while sauteing to promote even cooking. Ensure the heat is not too high for cooking the onions, as this could cause the onions to burn.
To caramelize onions, slice 2 onions, preferably a sweet variety. Melt 2 tablespoons butter in a large skillet over medium-low heat. Add the onions to the skillet, and cook them for 13 to 15 minutes, keeping the skillet covered. Cook the onions until tender, stirring periodically. Remove the cover from the skillet, and cook the onions for three to five minutes until each onion is a golden brown color.
Sauteed or caramelized onions can be used as a topping for a burger or sandwich or as an omelet filling. Combine sauteed onions with spinach to make a side dish, or use the onions to top a baked potato. For an appetizer, toast French bread, and spread it with goat cheese. Then, top the bread with caramelized onions and fresh thyme.