In a large saute pan over medium high heat, melt the butter. Add shallots and cook for about 1 minute. Stir in carrots, rosemary, salt, and pepper. Toss to coat well. Pour in chicken broth and bring mixture to a boil. Cover pan and reduce heat to a simmer. Cook for about 7-10 minutes, or until carrots are barely tender.
Increase the heat back to medium high and add in brussels sprouts. Cook, uncovered, for about 5 minutes, or until the brussels sprouts are starting to soften, but still crisp, and liquid is almost entirely absorbed.
Serve as a side dish to any grilled, roasted, or pan-seared item! So practically anything! Enjoy the holiday!