In a large skillet, heat the olive oil and butter over medium heat. Once melted, add the sliced onions.
Cook the onions slowly, stirring occasionally, until they are deep golden brown and caramelized, about 20-25 minutes. Reduce heat if necessary to prevent burning.
In a mixing bowl, whisk the eggs, salt, pepper, and thyme together until well combined.
Once the onions are caramelized, spread them evenly in the skillet. Pour the egg mixture over the onions, making sure they are evenly distributed.
Sprinkle the grated Gruyère cheese over the top of the egg mixture.
Cook on the stove for about 5 minutes, without stirring, until the edges start to set.
Preheat the broiler on your oven. Once the edges are set, transfer the skillet to the broiler and cook for an additional 2-3 minutes, or until the top is puffed and golden brown.
Remove from the oven, let it cool for a few minutes, then slice into wedges. Garnish with fresh chives if desired.