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To prep carrots, cut all but the last ½-inch of greens. Carefully scrub the carrots and around the carrot base to remove any dirt. Slice the carrots into halves or quarters lengthwise, and then slice into 4-inch long segments with dramatic diagonal cuts.
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Melt the butter in a 9- or 10-inch skillet with deep sides over medium heat. When butter is frothy and fragrant, add the carrots and cook, stirring often, for 7 minutes, or until carrots have softened a bit. Fold in the sugar, salt, pepper, broth, and radishes (if using). Cook carrots for another 7 - 10 minutes, or until the carrots are cooked through with a little snap in the middle and the glaze has cooked down. When the carrots are almost ready, fold in the lemon juice. Taste and add sea salt or pepper as needed.
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Meanwhile, toss the shallots in a small mixing bowl with the vinegar, lemon zest, and sea salt. Set aside for 5 - 10 minutes. Fold in the greens and olive oil. Add sea salt to taste.
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To serve, spoon carrots onto a platter and scrape any glaze from the pan on top. Add several spoonfuls of chimichurri on the side.