Step 1Cook the onion and olive oil in a small skillet over medium-high heat. Season with a pinch of salt and black pepper. Cook until the onion softens, about 10 minutes, then reduce the heat to medium, add the butter and continue to cook, stirring occasionally, until the onions are a deep golden brown, an additional 15 to 20 minutes. This makes about one-half cup.
Step 1Heat the garlic cloves in the olive oil in a small saucepan over medium heat until the garlic becomes soft and begins to brown. Remove from the heat and set aside to let cool. Refrigerate for storage.
Step 1As soon as the eggs are not runny, but not completely set, distribute the onions over top in clumps, scatter over the olives and garlic cloves, and crumble on the goat cheese. Guide the eggs from the pan, so it slips evenly onto the plate and sprinkle with thyme.