Tip:To make citrus segments, just slice the ends off the fruit with a small, sharp knife. Stand the fruit on one of its cut ends and slice off the skin in strips (try to get all the bitter white pith). Working over a bowl, cut the segments free from the membrane, letting each segment fall into the bowl as you go. Remove any seeds from the segments.
Heat the oven to 200°F. Melt 1-1/2 Tbs. of the butter in a 10- to 12-inch nonstick skillet over medium heat. Add the bread cubes to the pan, stirring and tossing almost constantly with a wooden spoon, until the cubes are golden brown and crisp, 3 to 5 minutes. Drain on paper towels and sprinkle with salt. Wipe out the skillet.
Segment the whole lemon. Cut each segment into four pieces. Transfer to a small bowl and add 1 Tbs. juice squeezed from the lemon half.
Season the sole fillets lightly on each side with salt and pepper and dredge the fillets in the flour, shaking off any excess. Heat the oil in the nonstick skillet over medium-high heat. Working in batches to avoid crowding the pan, cook the fillets until the first side is light golden, about 2 minutes. Using two spatulas to support the delicate fish, flip the fillets and cook until the second side is light golden and the fillets are opaque, another 2 minutes. Transfer to two dinner plates and keep warm in the oven.
Wipe out the skillet with a paper towel and add the remaining 3 Tbs. butter. Over medium-high heat, melt the butter and cook, swirling the pan until the butter turns a medium brown, about 2 minutes. Immediately remove the skillet from the heat and add the lemon segments and lemon juice, the capers, and the parsley; season to taste with salt and pepper. Drizzle the sauce over the fish and sprinkle on the croutons.
Serve with herbed rice pilaf.