Combine the flour, 2 teaspoons salt, and 1 teaspoon pepper in a shallow bowl. Remove the tenders on the underside of the chicken breasts and save for another use. (This makes the chicken breasts an even thickness so they cook evenly.) Heat the canola oil and 4 tablespoons of the butter together in a large (12-inch) cast-iron skillet over medium heat until the butter has melted and begins to sizzle.
Dredge the chicken in the flour mixture and shake off any excess. Place the chicken, skin side down, in the skillet and cook over medium heat without moving for 12 minutes. With the chicken still skin side down, place the skillet in the oven and roast for 10 minutes.
Return the skillet to the stovetop over medium heat (careful; the handle will be hot!). Turn the chicken skin side up, add the remaining 5 tablespoons of butter, and continue cooking for 3 to 5 minutes, until the butter is a nutty brown color and the chicken breasts are just cooked through (an instant-read thermometer will register 130 to 140 degrees F). Off the heat, sprinkle on the parsley.
Divide the Lemon Orzo with Feta among four dinner plates. Transfer the chicken to a cutting board and slice each breast crosswise in thick slices. Arrange one breast on each plate with the orzo. Spoon a few tablespoons of the brown butter over each chicken breast, squeeze on some lemon, and sprinkle with salt. Serve hot with a fresh wedge of lemon on each plate.
Lemon Orzo with Feta:
In a large saucepan, bring 2 quarts of water to a boil. Add 2 teaspoons salt and the orzo. Return the water to a boil and simmer uncovered for 9 to 11 minutes, until the orzo is al dente. Drain.
Transfer the orzo to a bowl and stir in the lemon zest, lemon juice, olive oil, dill, 2 teaspoons salt, and 1 teaspoon pepper. Fold in the feta and serve hot.