In a large bowl, combine 2 eggs, potatoes, parmesan, cheddar, chives and flour until well blended. Season with salt and pepper (just a pinch of each since your potatoes may have already been seasoned.)
Roll into 1 1/2 inch balls and chill until cold, approximately 2 hours.
Beat the remaining eggs in a bowl and place the breadcrumbs in a separate bowl.
Add the vegetable oil into a medium skillet so it’s approximately 2 inches deep and heat over medium-high until a pinch of breadcrumbs bubbles immediately when added.
Coat the mashed potato balls in the egg, then dredge in breadcrumbs to cover completely.
Working in batches, fry the croquettes until golden brown and crisp, 3–4 minutes. (Be sure to turn the croquettes so they brown evenly.)
Drain on paper towels and serve immediately with sour cream for dipping.
Potato Croquettes can be assembled up to 24 hours in advance. Cook right before serving.