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1. Adjust oven rack to lower-middle position and heat oven to 325 degrees. Heat 1 teaspoon oil in 12-inch heavy-bottomed ovensafe skillet over medium heat until shimmering. Add prosciutto and cook, stirring frequently, until just starting to brown, about 3 minutes. Add garlic slices and cook, stirring frequently, until garlic is golden brown, about 11/2 minutes. Using slotted spoon, transfer garlic and prosciutto to small bowl and set aside. Do not rinse pan.
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2. Increase heat to medium-high; add remaining 2 teaspoons oil and heat until just smoking. Pat chicken dry with paper towels and season with ground black pepper. Add chicken, skin side down, and cook without moving until well browned, 5 to 8 minutes. Using tongs, turn chicken and brown on second side, about 5 minutes longer. Transfer chicken to large plate.
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3. Remove all but 2 tablespoons fat from pan. Sprinkle flour over fat and cook, stirring constantly, for 1 minute. Slowly add wine and broth; bring to simmer, scraping bottom of pan with wooden spoon to loosen browned bits. Cook until liquid is slightly reduced, 3 minutes. Stir in cloves, rosemary stem, sage leaves, bay leaves, red pepper flakes, and reserved prosciutto and garlic. Nestle chicken into liquid, skin side up (skin should be above surface of liquid), and bake, uncovered, until meat offers no resistance when poked with fork but is not falling off bones, about 1 hour 15 minutes. (Check chicken after 15 minutes; broth should be barely bubbling. If bubbling vigorously, reduce oven temperature to 300 degrees.)
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4. Using tongs, transfer chicken to serving platter and tent with foil. Remove and discard sage leaves, rosemary stem, cloves, and bay leaves. Place skillet over high heat and bring sauce to boil. Cook until sauce is reduced to 11/4 cups, 2 to 5 minutes. Off heat, stir in minced rosemary, lemon juice, and butter. Season to taste with salt and pepper. Pour sauce around chicken and serve.
