Make the French onion dip: Add the onions to a large sauté pan over low heat and season with salt, pepper, and the garlic powder. Slowly cook the onions, stirring occasionally, until caramelized, about 45 minutes. Carefully pour in the white wine and let evaporate to cook out the alcohol, 2–3 minutes. Transfer half of the cooked onions to a medium bowl and set aside.
Remove the pan from the heat and stir in the cream cheese until fully incorporated. Add the Parmesan, mayonnaise, and sour cream and stir to combine. Transfer to a bowl and garnish with the chives. Serve the warm or at room temperature. (If served cold, the dip will be slightly thicker.)
Make the wontons: To the bowl with the reserved caramelized onions, add the cream cheese and stir with a rubber spatula until fully incorporated. Add the chives and stir to combine. Transfer to the refrigerator to set while you heat the oil.
Add the oil to a heavy bottomed pot and heat over medium heat until it reaches 350°F (180°C).
Fill each wonton wrapper with 1 teaspoon of the chilled onion dip. Brush the edges of the wrappers with the beaten egg. then bring the opposite corners up and over the filling to meet in the center to create an X shape. Pinch the edges together to seal well.
Working in batches, fry the wontons in the hot oil until golden brown, 3–4 minutes. Transfer to a paper towel-lined plate to drain.
Transfer the wontons to a platter and serve immediately with the French onion dip and chips or pretzels for dipping.