Place the onion, parsley leaves, parsley stalks, mustard, salt and pepper into the food processor and pulse until the onion is minced but not pureed. Transfer to a large bowl and set aside.
Process the meat in the processor; 20 fast pulses. Transfer to the bowl with the onion mixture, add the rice and stir until well mixed. Form into 4 patties of equal size. This step will be easier if you chill the chicken-rice mixture in the refrigerator for an hour.
Heat the oil in a large skillet over medium heat. Add the chicken burgers and cook 4 minutes on each side.
To make the Horseradish-Wine Sauce, deglaze the skillet with the wine, scraping to loosen up the bits of pan residue into the liquid. Strain into a small bowl. Stir in the yogurt and horseradish and set aside.