Preheat oven to 425 degrees Fahrenheit.
Combine the brown sugar, paprika, Italian seasoning, 1 teaspoon salt, garlic powder, pepper, and chili powder in a small bowl and mix until combined.
Place the potatoes in a 9x13" baking dish and drizzle with 1 tablespoon of oil. Toss to coat. Sprinkle 1 ½ teaspoons of the seasoning mix over the potatoes.
Place the potatoes in the oven and bake, uncovered, for 30 minutes.
Drizzle the remaining ½ tablespoon of olive oil over the chicken and season front and back with the remainder of the seasoning mix. Set aside until the potatoes are finished cooking.
After 30 minutes remove the potatoes from the oven and arrange the seasoned chicken breasts on top of the potatoes.
Return the pan to the oven and continue to cook for 20 minutes, or until the chicken is cooked through and has an internal temperature of 165 degrees F.
In a small saucepan, combine the butter, lemon juice, garlic, remaining ¼ teaspoon salt, and dried dill weed over medium-low heat. Stir and cook until the garlic is softened and very fragrant. Remove from the heat and set aside.
When the chicken and potatoes are done, remove the pan from the oven and pour the garlic butter sauce over the top of the chicken and potatoes. Allow the chicken to rest for 5 minutes and serve.