Arrange a rack in the lower third of the oven. Preheat oven to 400°F.
Heat 1 tablespoon of olive oil in a large (12-14 inch) cast iron skillet on the stovetop over medium-high heat.
Pat the chicken dry with a paper towel; season with salt and pepper to taste.
Place the chicken skin-side down in the hot skillet; add the rosemary sprigs. Cook until the chicken is golden and crisp, about 10 minutes. Flip the chicken and continue cooking until it’s golden on both sides, about 4-5 more minutes.
Transfer the chicken to a plate or tray; discard rosemary.
Pour the drippings from the skillet into a small bowl; reserve for later.
Add about ¼ cup of water to the skillet and stir with a wooden spoon to loosen browned bits from the bottom of the pan. Discard the liquid; wipe the skillet clean.
Heat the remaining 3 tablespoons of olive oil in the skillet over medium-high heat. Add the sweet potatoes and cook for 2 minutes. Add the Brussels sprouts and shallots. Season the vegetables with salt and pepper, to taste. Cook until charred, about 5 minutes. Remove the skillet from the heat.
To the small bowl with the reserved drippings, add the garlic, maple syrup, melted butter, Dijon mustard, cider vinegar, and Worcestershire sauce; whisk to combine.
Place the chicken skin-side up on top of the vegetables in the skillet. Pour the maple sauce evenly over the chicken and vegetables.
Transfer the skillet to the preheated oven. Bake until the vegetables are tender and the chicken reaches an internal temperature of 160 - 165°F, about 15-20 minutes (mine are typically done in 18 minutes).
Remove the skillet from the oven and let the chicken rest for about 5-10 minutes before serving. Taste and season with additional salt and pepper. Garnish with chopped fresh herbs, if desired.