Heat 2 tablespoons oil in a large skillet over medium heat. Add onions and season with salt and pepper. Cook, stirring occasionally, until onions are a deep golden brown, 35 to 40 minutes.
Meanwhile, heat remaining 2 tablespoons oil in a small saucepan over medium-high heat. Add shallots and cook, stirring occasionally, until golden brown, 3 to 4 minutes. Transfer to a paper towel-lined plate with a slotted spoon; cool completely.
Combine sour cream and onions in a bowl. Season with salt and pepper. Top with crispy shallots and chives. Serve with pretzels and chips.
