In a large skillet, heat 1 tablespoon olive oil over medium-low heat. Add onions, season with coarse salt. Cook, stirring frequently, until onions are caramelized, 35 to 40 minutes.
In a medium saucepan, combine potatoes, garlic, and enough water to cover by 1 inch. Bring to a boil; add salt. Cover; reduce heat, and simmer until tender, 15 to 20 minutes. Drain, reserving 1 cup liquid.
Return potatoes and garlic to pan. Add remaining 2 tablespoons olive oil; season with salt. Mash until smooth, adding reserved liquid as needed. Stir in onions, and serve immediately.