Combine the vinegar and honey in a liquid measuring cup and whisk
Cut 1 TBSP of the butter into 4 pieces and refrigerate
In a 12" skillet, heat the remaining 1 TBSP of butter with the olive oil over medium 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 the carrots is lightly browned in spots, 4-5 minutes
Using tongs, stir and flip the carrots and then leave undisturbed for 1-2 minutes to brown. Continue cooking, occasionally stirring and flipping, until most of the carrots are a bit browned in places and starting to feel tender 3-5 minutes
Reduce heat 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 TBSP of the broth has evaporated about 2 minutes
Uncover, reduce heat to medium low, and add the pomegranate mixture(rewhisk if necessary) and the cayenne
Cook, stirring gently, until the mixture reduces and becomes slightly glazy, about 1 minute. Take the pan off of the heat and add the chilled butter and gently toss until the butte has melted, 30 seconds to 1 minute. Season to taste with salt and stir in 2/3 of the mint. Use the the remaining mint for garnish
