In a 12-inch nonstick skillet, sprinkle 1 1/2 teaspoons curry powder; over medium heat, stir until fragrant, about 2 seconds.
Add carrots, salt, 1 tablespoon sugar, and chicken broth to the skillet and bring to a boil, covered, over medium-high heat; reduce heat to medium and simmer, stirring occasionally, until carrots are almost tender when poked with tip of paring knife, about 5 minutes.
Uncover, increase heat to high, and simmer rapidly, stirring occasionally, until liquid is reduced to about 2 tablespoons, 1 to 2 minutes.
Add butter, currants, and remaining 2 tablespoons sugar to skillet; toss carrots to coat and cook, stirring frequently, until carrots are completely tender and glaze is light gold, about 3 minutes.
Turn off heat, add lemon juice; toss to coat. Transfer carrots to serving dish, scraping glaze from pan. Season to taste with pepper and serve immediately.