1. Place 2 to 3 tablespoons extra virgin olive oil (or half oil, half unsalted butter) in a heavy 10-inch skillet or pot over very low heat. Add 1 1/2 to 2 pounds sliced onions (1 pound of raw onions yields about 1 cup caramelized). The onions will be 2 to 3 inches deep; stir to coat with oil. Cover tightly and cook over very low heat, stirring occasionally, for 45 to 60 minutes, until completely softened and beginning to brown.
VARIATION: Quick Caramelized Onions
Cook sliced onions as in the preceding recipe, only increase the heat to medium or medium-high. Cover and stir often until the onions are softened and beginning to brown. Remove the lid; sprinkle the onions with 1 to 2 tablespoons granulated sugar, stirring to combine. Continue to cook until the onions are caramelized as desired.
VARIATION: Caramelized Onions Balsamico
Stir 1 to 2 tablespoons good quality balsamic vinegar into the caramelized onions; cook until liquid evaporates.