To a medium skillet set over medium heat, add the oil and three quarters of the onions and cook, stirring occasionally, until soft and beginning to brown, about 14 minutes. Set aside to cool.
In a large bowl, cover the matzos with cold water and set aside for 10 minutes. Drain, use your hands to squeeze out as much liquid as possible, and return to the bowl.
In a food processor, pulse the carrot and remaining onion until very finely chopped, then add the matzo and pulse to combine. Scrape into the bowl and set aside.
Position a rack in the center of the oven and preheat to 425ºF. To the empty food processor, add the whitefish and onions and pulse to a paste (the fish chunks should be no larger than a grain of rice). Scrape into the bowl with the matzo mixture. To the food processor, add the carp, salt, sugar, and white pepper and pulse to a paste. Scrape into the bowl with the whitefish, then add the eggs. Using your hands or a spoon, mix until homogeneous. (If the mixture is too loose to form, cover with plastic and refrigerate for 1 hour.)
Using two large spoons or wetted hands, shape about 2 tablespoons of the mixture into a quenelle or oblong ball and transfer to a greased 9-by-13-inch pan. Brush lightly with oil. Repeat with the rest of the mixture, then bake until just beginning to brown, about 21 minutes.
Meanwhile, in a small pot, bring the stock and sliced carrot to a simmer, then pour over the fish to come two thirds of the way up the quenelles. Loosely cover with foil and turn the oven down to 325ºF. Bake, turning the balls over halfway through cooking, until cooked through and moist, about 45 minutes.
Remove the pan from the oven and allow the fish to cool in the liquid, about 3 hours. Cover with plastic and refrigerate until thoroughly chilled, at least 3 hours.
To serve, cover the bottom of a large platter with parsley if desired and scatter with the radishes, then arrange the gefilte fish on top. Top each fish with a carrot slice and parsley sprig and serve cold with horseradish on the side.