1. Trim tops of carrots to 1-inch; peel. Set aside.
2. Melt butter in an 8-quart pressure cooker over medium-high heat. Add oil to cooker; swirl to coat. Sprinkle chicken evenly with salt and pepper. Place flour in a shallow dish. Dredge chicken in flour. Place 2 chicken leg quarters, flesh side down, in cooker; sauté 5 minutes or until browned. Set aside. Repeat procedure with remaining 2 chicken leg quarters. Place mushrooms in cooker; sauté 4 minutes or until liquid evaporates. Remove mushrooms from cooker using a slotted spoon; set aside.
3. Stir in wine, scraping cooker to loosen browned bits. Bring to a boil; cook 30 seconds. Tie thyme and sage sprigs together with twine. Add chicken, herb sprigs, and broth to cooker. Close lid securely; bring to high pressure over high heat. Reduce heat to medium or level needed to maintain high pressure; cook 6 minutes. Remove from heat; release pressure through steam vent, or place cooker under cold running water to release pressure. Remove lid.
4. Add carrots, mushrooms, and onions to cooker; close lid securely. Return cooker to high pressure and immediately remove from heat; release pressure through steam vent, or place cooker under cold running water to release pressure. Remove lid. Transfer chicken to a platter. Remove vegetables from cooker using a slotted spoon; arrange on platter with chicken. Cover and keep warm. Strain cooking liquid through a cheesecloth-lined sieve into a large bowl; discard solids. Transfer liquid to a large, wide skillet over medium-high heat; bring to a boil. Cook until reduced to 1 cup (about 12 minutes). Stir chopped thyme and sage into sauce. Serve with chicken and vegetables.
Wine note: Chardonnay loves sweet root vegetables (especially onions) with chicken. For this dish, deliciously browned and laced with mushrooms and fresh herbs, the chardonnay needs to have its sweet fruit balanced with an earthy, minerally side to work with all the layers. Iron Horse 2008 Chardonnay (Green Valley of Russian River Valley, $27) is perfect--a bushelful of juicy apples balanced with creamy lemon, a hint of gravelly minerality, and the barest touch of toast. --Sara Schneider