Combine the maple syrup, lime juice, and 1 Tbs. water in a small dish and set near the stove. Set a shallow serving dish near the stove, too.
In a 10-inch straight-sided sauté pan, heat 1 Tbs. of the butter with the olive oil over medium-high heat. When the butter is melted, add the carrots and season with 3/4 tsp. salt. Toss with tongs to coat well. Cook, gently tossing occasionally at first and then more frequently, until most of the carrots are well browned and tender when pierced with a fork, 6 to 9 minutes (if the carrots aren’t fully tender but look like they’re burning, reduce the heat to medium).
Reduce the heat to low, add the remaining 1 Tbs. butter and the ginger and cook, stirring and scraping the bottom of the pan with a heatproof rubber spatula, until the butter has melted and the ginger is fragrant, 15 to 20 seconds. Carefully add the maple syrup mixture and cook, stirring, until the liquid reduces to a glazey consistency that coats the carrots, 15 to 20 seconds.
Immediately transfer the carrots to the serving dish, scraping the pan with the spatula to get all of the gingery sauce. Let sit for a few minutes and then serve warm.