Pound 2 pounds medium boneless, skinless chicken thighs a few pieces at a time: Place in a large zip-top bag and pound with a rolling pin or the flat side of a meat mallet until about ¼-inch thick. Pat the chicken dry with paper towels, then season all over with 1 teaspoon kosher salt and ½ teaspoon coarsely ground black pepper.
Place ½ cup all-purpose flour in a large, shallow bowl. Place 2 large eggs and ¼ cup sour cream in a second large, shallow bowl and whisk with a fork until broken up with no streaks of egg white. Place 2 cups panko breadcrumbs in a third large, shallow bowl.
Working with 1 piece of chicken at a time, dip in the flour mixture, then the egg-sour cream mixture, and finally the panko mixture until evenly coated, gently pressing the breadcrumbs into the chicken so they stick. Place on a baking sheet in a single layer.
Heat 1 cup neutral oil in a medium Dutch oven (about 5 ½ quarts) over medium-high heat until 350ºF. Working in 3 to 4 batches, add the chicken to the hot oil in a single layer and fry until cooked through and light golden-brown, 1 ½ to 3 minutes per side. Transfer to a paper towel-lined plate or wire rack set over a baking sheet. Skim off any breadcrumbs in the oil between each batch. Serve with tonkatsu sauce, thinly sliced green cabbage, and rice if desired.
