1. Preheat oven to 325 degrees.

2. In a large heavy skillet, heat olive oil over medium-high heat. Season all the pork with salt and pepper. Add the pork, in batches, and brown on all sides, about 12 minutes per batch. Transfer pork to a rack set over a baking sheet.

3. Add onions and carrots to skillet and cook until well colored and almost caramelized, about 15 minutes. Add the bouquet garni and 1 clove garlic. Remove skillet from heat and carefully add bourbon. Return to heat and simmer until liquid is reduced by three-quarters. Add wine and reduce by half.

4. Transfer vegetables and braising liquid to a large enameled cast-iron casserole or Dutch oven. Add pork, remaining clove garlic, and chicken broth. Bring to a simmer over medium-high heat, then reduce heat to low. Cover the surface of the liquid with a sheet of parchment paper and cover the pot with a lid. Place in the oven and cook until the pork is tender, about 1 1/2 hours.

5. Using a slotted spoon, transfer pork to a rack set over a baking sheet. Strain remaining liquid into a saucepan. Set saucepan halfway on burner and bring to a boil over high heat, skimming fat from top, until liquid is reduced by half.

6. Return pork to sauce and cook until heated through. Add butter and swirl in pan until melted; season with salt and pepper. Serve immediately.