Tuck wing tips under chicken. Starting at neck opening, gently loosen skin from breast of chicken. Slide 2 sage leaves, 1 thyme sprig, and 1 rosemary sprig under skin of each chicken breast half. Sprinkle 1/4 teaspoon coarse salt inside chicken cavity and insert 1 thyme sprig and 1 rosemary sprig into cavity. Sprinkle outside of chicken all over with 2 teaspoons coarse salt. Place chicken, breast side up, on plate. Let stand uncovered at room temperature 2 hours or cover and chill overnight.
Position oven rack in lower third of oven; preheat to 450°F. Coat large rimmed baking sheet with nonstick spray. Place chicken in center of prepared sheet. Place beets in medium bowl. Add 2 teaspoons olive oil, 1/2 tablespoon chopped thyme, 1/2 tablespoon chopped rosemary, and 1 teaspoon chopped sage; sprinkle with salt and pepper and toss to coat. Place parsnips and Jerusalem artichokes in another medium bowl. Add remaining 2 teaspoons olive oil, 1/2 tablespoon chopped thyme, 1/2 tablespoon chopped rosemary, and 1 teaspoon chopped sage; sprinkle with salt and pepper and toss to coat. Arrange all vegetables around chicken on sheet. Roast chicken and vegetables 20 minutes. Reduce heat to 375°F. Roast until instant-read thermometer inserted into thickest part of chicken thigh registers 165°F and vegetables are tender, turning vegetables occasionally, about 50 minutes longer.
Using tongs, tilt chicken to allow juices to drain from main cavity onto baking sheet. Transfer chicken to platter. Arrange vegetables around chicken. Place baking sheet over 2 burners. Add beer to baking sheet and boil until pan juices are slightly reduced, scraping up browned bits, 3 to 4 minutes. Season juices to taste with salt and pepper. Transfer juices to small pitcher. Serve chicken and vegetables, passing pan juices alongside.