Cook the rice and keep it warm. Whisk the dressing ingredients in a small bowl and set it aside.
Pat the duck very dry. Score the skin in a crosshatch pattern with a sharp knife, cutting through the skin and fat but not into the meat. Season both sides with the salt and pepper.
Place the breasts skin-side down in a cold, large skillet. Turn the heat to medium-low. Cook for 12 to 14 minutes, until a lot of fat has melted into the pan, the skin is deep golden brown, and it sizzles steadily rather than spitting. Pour off the fat into a heatproof jar twice during cooking, so the skin fries rather than boils.
Flip the breasts and cook the meat side for 5 to 6 minutes, until a thermometer in the thickest part reads 74°C (165°F). Transfer the duck to a cutting board, skin-side up, and rest for 8 minutes.
While the duck rests, pour off all but 1 tablespoon of fat from the skillet and heat it over medium-high heat. Add the artichokes cut-side down and cook for 4 to 5 minutes without moving, until deeply golden at the edges. Add the garlic for the last 30 seconds, until fragrant. Season with a pinch of salt.
Slice the duck thinly against the grain, keeping the crispy skin on each slice.
Divide the rice among four bowls. Top with arugula, artichokes, and sliced duck. Drizzle with the dressing, and scatter the almonds and Parmesan over the top. Serve right away so the skin stays crisp.