Whisk the mustard and 2 cups water in a large bowl or wide pot until smooth; add the thyme sprigs. Add the chicken and onion, turning to coat well. Firmly press down so the chicken is just covered in the marinade. Cover and refrigerate 2 to 4 hours.
Fill two 12-inch cast-iron skillets with about 2 1/2 inches oil each. Heat over medium heat until a deep-fry thermometer registers 350 degrees F.
While the oil heats, make the breading: Put 3 cups flour in a large bowl. Remove 1/3 cup of the chicken marinade and add to the flour; stir until clumpy. Whisk in the remaining 2 1/2 cups flour, the baking powder, paprika, dill, cayenne, 1 tablespoon salt and 2 teaspoons black pepper. Working with a few pieces at a time, remove the chicken and onion from the marinade, letting the excess drip off, and dredge in the flour mixture, pressing with your hands to coat. Transfer to a rack set over a rimmed baking sheet.
Carefully add about 5 pieces chicken to each skillet, then adjust the heat to keep the oil temperature between 300 degrees F and 325 degrees F. Fry, turning the chicken occasionally, until golden and crisp on the outside and no longer pink inside, 15 to 16 minutes. (A thermometer inserted into the chicken should register 160 degrees F for breasts and 170 degrees F for legs and thighs.) Remove with tongs and transfer to a clean rack set over a rimmed baking sheet; season with salt. Return the oil to 350 degrees F and fry the remaining chicken. Fry the onion 2 to 3 minutes, stirring occasionally. Transfer the chicken and onion to a platter.