Whisk together ½ cup olive oil, apple cider vinegar, sugar, Dijon, salt, and pepper. Set the vinaigrette aside.
Cook the white rice and wild rice in two separate pots, following the package directions. Once cooked, let each rest covered for 5 minutes, fluff with a fork, and transfer to a large serving bowl.
While the rice cooks, heat 1 tablespoon olive oil in a large 15-inch cast-iron or other heavy-bottomed skillet over medium-high heat. Add the chicken and season with 1 teaspoon salt and ½ teaspoon pepper.
Cook on one side until the chicken begins to turn opaque. Flip the chicken, add the cranberries, and cook for 1-2 minutes more, until the chicken is nearly cooked through. Do not overcook; it will finish cooking in step 10.
Transfer the chicken, cranberries, and any accumulated juices to a bowl and set aside.
Heat 1 tablespoon olive oil in the same skillet over medium-high heat and add the mushrooms and 1 teaspoon salt. Don't worry if the skillet seems crowded; the mushrooms will shrink considerably as they cook.
The mushrooms will release liquid as they cook. Once the liquid has evaporated, add the remaining 1 tablespoon olive oil, the onions, and dried thyme. Continue cooking, stirring occasionally, until the onions are translucent (about 3-4 minutes).
Add the garlic and continue cooking for 2 minutes, until fragrant and the mushrooms are deeply browned.
Pour in the dry sherry and scrape up any browned bits from the bottom of the skillet. Simmer until the sherry has reduced by about half.
Return the chicken, cranberries, and reserved juices to the skillet. Cook until the chicken reaches 165°F and is warmed through. Season with additional salt and pepper to taste.
Add the chicken and mushroom mixture, along with any juices, to the bowl of rice and stir to combine.
Pour 3/4 of the vinaigrette over the rice and toss gently until evenly coated. Add more vinaigrette to taste. If you expect to have leftovers, reserve a little vinaigrette for serving the next day.
Just before serving, add the arugula and toss gently. Garnish with cranberries if desired.