Lay the duck breasts skin side up on a cutting board and pat them very dry with paper towels, then season both sides evenly with kosher salt and black pepper.

Use a sharp paring knife on the cutting board to lightly score the skin in a crosshatch pattern, cutting through the fat but not into the meat, so the fat can render out.

Set a cast iron skillet on the stove and place the duck breasts in skin side down with tongs while the pan is still cool. Turn the burner to medium heat so the fat slowly melts and the skin starts to sizzle.

Let the duck cook skin side down until the skin is deep golden and crisp and you see a good layer of rendered fat in the skillet, then flip the breasts with tongs and cook the meat side just a few minutes, until the duck feels slightly springy when pressed with a finger.

Move the duck breasts to a clean cutting board with tongs and let them rest while you make the sauce, leaving a thin layer of duck fat in the skillet and pouring off any excess if there is too much.

With the cast iron skillet back on medium heat, add the blueberries, chicken broth, balsamic vinegar, honey, lemon juice, and a pinch of kosher salt, then stir with a wooden spoon to loosen up the browned bits from the bottom.

Let the sauce simmer a few minutes, stirring with the wooden spoon, until the blueberries soften and burst and the liquid thickens into a glossy, spoon-coating sauce; if it gets too thick, splash in a little extra broth.

Slice the rested duck breasts on the cutting board with a sharp knife, fan the slices onto plates, and spoon the warm blueberry sauce over the top with the wooden spoon, making sure some berries land on each serving.