Preheat the oven to 425˚. Bring a medium saucepan of salted water to a boil. Season the chicken with salt, pepper, the paprika and brown sugar. Heat 2 tablespoons vegetable oil in a large ovenproof skillet over medium-high heat. Add the chicken, reduce the heat to medium and cook until browned, 6 minutes. Flip the chicken; transfer the skillet to the oven. Bake until cooked through, 10 minutes. Remove the chicken to a plate and cover with foil.
Pour off the juices from the skillet; reserve. Add the remaining 1 tablespoon oil to the skillet and set over medium-high heat. Add the shallots and cook, undisturbed, until browned, 4 to 5 minutes. Toss; cook until tender, 2 more minutes. Add the dates and vinegar; cook until the vinegar is reduced, 2 minutes. Add the chicken broth, bring to a simmer and cook until reduced, 3 to 4 minutes. Add the reserved skillet juices and any juices from the plate of chicken.
Meanwhile, cook the noodles in the boiling water as the label directs. Reserve 1/2 cup cooking water, then drain the noodles and return to the pan. Add the butter and 1/4 cup of the reserved cooking water. Toss until coated, adding more cooking water as needed. Season with salt and pepper and stir in 3 tablespoons parsley.
Slice the chicken and top with the sauce. Serve with the noodles; sprinkle with the remaining 1 tablespoon parsley.