To make the caramelized onions, heat 2 tablespoons of butter and the olive oil in a large sauté pan over medium heat. Add the sliced onions and stir them to coat with the fat. Add 1/4 teaspoon salt and the sugar and cook, stirring frequently, until onions are golden brown and caramelized, about 30-40 minutes. If the onions get too dry, add a small amount of water to deglaze the pan. Set aside.
Place the potatoes in a large pot of cold, salted water. Bring to a boil and cook until fork tender.
While the potatoes are cooking, heat the half and half, garlic, bay leaf, and remaining 2 tablespoons of butter in a small saucepan over medium heat.
Drain the potatoes and return them to the hot, dry pot. Stir them over low heat for 2 minutes until they are dry. Pass the potatoes through a ricer into a large bowl. Gently stir in the hot cream mixture a little at a time until the potatoes are smooth and creamy (discard the garlic and bay leaf). You may not need to use all of the liquid. Set aside a small amount of the caramelized onions for garnish and stir the remaining onions into the mashed potatoes. Crumble the goat cheese into the potatoes and stir to combine well. Season with salt and black pepper to taste.
Spoon the mashed potatoes into a serving bowl and garnish with the remaining caramelized onions. Serve hot.