Melt butter in a heavy skillet over medium-high heat until foaming.
Add carrots and salt; cook and stir in hot butter until liquid evaporates and carrots begin to brown around the edges, 5 to 6 minutes.
Reduce heat to medium-low and pour in bourbon carefully.
Cook and stir until bourbon is almost evaporated, about 2 minutes.
Sprinkle in brown sugar; cook and stir until carrots are almost cooked through, about 5 minutes more.
Increase heat to medium-high and cook until glaze thickens, 15 to 30 seconds.
Season with ground black pepper and cayenne pepper.
Transfer carrots to a serving dish and garnish with thyme leaves.
