1/4 cup pure pomegranate juice
1 Tbs. balsamic vinegar
2 tsp. honey
1 oz. (2 Tbs.) unsalted butter
2 Tbs. extra-virgin olive oil
2 lb. carrots, trimmed, peeled, and cut into sticks about 2 inches long and 3/8 inch wide
Kosher salt
1/3 cup lower-salt chicken broth
1/8 tsp. cayenne
2 Tbs. lightly packed thinly sliced fresh mint

1. The carrots can be cut up 6 hours ahead. Cut the carrots into 2 inch pieces. Cut those pieces down the middle. Place the flat side on the cutting board and cut into 3/8 inch wide strips.
2. Combine the juice, vinegar, and honey in a liquid measuring cup and whisk. Cut 1 Tbs. of the butter into 4 pieces and refrigerate.
3. In a 12-inch skillet, heat the remaining 1 Tbs. butter with the olive oil over medium-high heat. When the butter has melted, add the carrots and 1-1/2 tsp. salt and toss well to coat. Cook without stirring until the bottom layer of carrots is lightly browned in spots, 4 to 5 minutes. Using tongs, stir and flip the carrots and then leave undisturbed for 1 to 2 minutes to brown. Continue cooking, occasionally stirring and flipping, until most of the carrots are a bit browned in places and are starting to feel tender, an additional 3 to 5 minutes. Reduce the heat to medium if the bottom of the pan begins to brown too much.
4. Carefully add the chicken broth, cover quickly, and cook until all but about 1 Tbs. of the broth has evaporated, about 2 minutes. Uncover, reduce the heat to medium low, and add the pomegranate mixture (re-whisk, if necessary) and the cayenne. Cook, stirring gently, until the mixture reduces and becomes slightly glazy, about 1 minute.
5. Take the pan off the heat, add the chilled butter, and gently toss with a heatproof spatula until the butter has melted, 30 seconds to 1 minute.
6. Season to taste with salt and stir in about two-thirds of the mint. Serve in a warm shallow bowl or on a platter, garnished with the remaining mint.

Serves 6 to 8.