In a shallow dish, season the sole with salt and black pepper and cover with the milk. To another shallow dish, add the flour and season generously with salt and black pepper.
Slice the top and bottom off the lemon. Using a sharp knife, slice down the sides of the lemon to remove and discard the zest and pith. Slice between the membranes to remove the segments, then cut the segments into ½-inch pieces and set aside.
To a skillet set over medium-high heat, add the clarified butter. Remove the fillets from the milk and tuck under the thin, tapered ends to create an even thickness. When the butter is hot and shimmering, dredge the folded fish in the flour, shaking off any excess, then cook, turning once, until golden brown, about 2 minutes per side. Transfer to a plate and tent with foil.
To the skillet, add the unsalted butter and cook, stirring occasionally, until it’s deep brown and smells nutty, about 3 minutes. Working quickly, remove from the heat and stir in the capers, parsley, and half of the lemon pieces (reserve the other half for another use). Spoon the sauce over the sole and serve immediately.