In a large cast-iron pan over medium heat, add the butter and heat until melted. Add the onions and 1/4 teaspoon salt. Cook the onions until they are caramelized, about 20 minutes. Deglaze with the vermouth and reduce until dry, about 2 minutes. Remove from the heat and set aside to cool.
Meanwhile, preheat a fryer or a large Dutch oven filled with 3 inches oil to 350 degrees F. Line a sheet tray with a wire rack. Fill a large pot with salted water and bring to a boil.
Add the sour cream, cream cheese, garlic powder and onion powder to a food processor, then add the cooled onions. Process until combined and smooth. Taste and adjust the seasoning as needed. Pour into a serving dish and refrigerate; stir again before serving.
Add the fingerling potatoes to the boiling water and cook for 12 minutes. Strain and set aside on a sheet tray or large plate until they are just cool enough to handle. Gently crush the potatoes, but still hold their shape. Fry until crispy, 3 to 5 minutes. Drain on the wire rack and season with salt.
Serve the crispy potatoes on a platter with the onion dip and caviar on the side.