Roast drumsticks for approximately 30 to 40 minutes in an oven that’s preheated to 425 F. To make a one-dish meal, roast chicken drumsticks on a bed of sliced or cubed potatoes and flavorful vegetables such as chopped kale, sliced onions and minced garlic.ven-frying chicken works for skin-on or skinless drumsticks. Preheat your oven to 400 F. Season the drumsticks with salt and pepper. Place breadcrumbs or panko bread crumbs in a shallow bowl. Season the breadcrumbs, if desired, with your preferred seasonings, such as garlic and paprika.