Bring a large pot of salted water to a boil over high heat.
In a medium, heavy saucepan over high heat, combine 1 cup of the bourbon, the corn syrup, brown sugar, butter, and mint. Bring to a boil, stirring until the sugar dissolves. Continue boiling until the syrup is reduced and thickened enough to coat the back of a spoon, about 8 minutes, stirring occasionally. Set a strainer over a measuring cup or small bowl and strain the glaze, pressing down to release all of the glaze (you should have about ¾ cup). Mix in the remaining 1 tablespoon bourbon, the vinegar, ⅛ teaspoon salt, and ⅛ teaspoon pepper. Put half of the glaze into a small saucepan.
Cut the yams diagonally into ½-inch slices, each 3 to 4 inches long. Cook the yams in the pot of boiling water until tender but not falling apart, 7 to 8 minutes. Drain thoroughly. Brush the yams generously on both sides with the remaining glaze and season each with a pinch of salt and pepper.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Brush the cooking grates clean. Grill the yam slices over direct medium heat, with the lid closed as much as possible, until well marked and heated through, 4 to 5 minutes, turning once. Transfer to a platter. Warm the glaze in the saucepan over medium heat. Drizzle the glaze over the yams and top with the mint. Serve warm.
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