Combine the juice, vinegar, and honey in a liquid measuring cup and whisk. Cut 1 Tbs. of the butter into 4 pieces and refrigerate.
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.
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. 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. 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.
Cut the carrots and prepare the rest of your ingredients up to 6 hours ahead. Cook the dish just before serving.