The night before serving, slice the potatoes 1/4-inch thick and transfer to a bowl of cold water. Refrigerate up to 24 hours, changing the water a few times. (This makes the potatoes extra crispy when fried.)
Melt 1 tablespoon butter in a large pot over medium heat. Add the red onion, white onion, shallots, thyme, 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring, until the onions are soft, 30 minutes. Stir in the brown sugar, brandy and Worcestershire sauce and cook until most of the liquid has evaporated, about 10 minutes. Remove from the heat and stir in the vinegar; transfer to a bowl and set aside.
Combine 3 teaspoons crushed peppercorns with 1 teaspoon salt in a large bowl. Add the steak and toss to coat, pressing the peppercorns into the meat. Melt the remaining 1 tablespoon butter in a large skillet over medium-high heat and cook until it just starts browning. Add the steak; cook, turning, until browned on all sides, about 6 minutes. Remove to a cutting board and let rest 10 minutes.
Meanwhile, drain the potatoes and pat dry. Transfer to a large pot; cover with 2 inches of vegetable oil. Cook over medium-high heat, gently stirring occasionally, until golden brown, about 20 minutes. Remove with a slotted spoon and transfer to a rack set on a baking sheet; season with salt. (The potatoes can be fried up to 2 hours ahead; reheat in a 300 degrees F oven for 5 minutes before assembling.)
Thinly slice the steak. Spread some of the onion mixture on each potato; top with a slice of steak and a dollop of sour cream. Sprinkle with the remaining 1 teaspoon crushed peppercorns.