Trim the tops and tails from the carrots and peel them; you should have about 8 oz. trimmed carrots. Cut them in half crosswise and then cut the thicker end in half lengthwise to get pieces of about the same width, no more than 3/4 inch (the length can vary). In a small bowl, combine the lime juice, lime zest, and brown sugar; set aside. Heat 1 Tbs. of the olive oil and 2 teaspoons of the butter in a 10-inch straight-sided sauté pan over medium-high heat. When the milk solids in the butter are just beginning to turn a nutty brown, add the carrots and 1/4 tsp. salt. Toss well with tongs and then arrange the carrots in one layer. Cook without stirring until the bottoms are nicely browned, 3 to 4 min. Toss and turn over, and cook for another 2 min. to lightly brown another side. Transfer the carrots to a plate with tongs
Heat the remaining 1 Tbs. olive oil in the pan. Add the bell pepper, red onion, and a pinch of salt and sauté until browned, 3 to 5 min. Add the ginger and jalapeño, toss, and sauté for another 1 min. Return the carrots to the pan, stir, and pour in the chicken broth. Immediately cover the pan and simmer until the liquid is almost completely reduced, about 2 min.
Uncover the pan, remove it from the heat, and add the lime juice mixture, the remaining 1 tsp. butter, and the cilantro. Toss to combine well, scraping any browned bits from the bottom of the pan with a heatproof spatula or a wooden spoon. Serve right away as individual servings or pour and scrape the contents of the pan onto a small platter and serve family style.