Heat 2 tablespoons olive oil in a large skillet over medium-high. Sprinkle lamb with 2 ½ teaspoons salt and 1 teaspoon pepper. Cook until browned, about 3 minutes per side. Remove lamb from skillet. Add onions, carrots, and garlic; cook, stirring occasionally, until slightly softened, about 4 minutes. Add ½ cup dry red wine; cook, until mostly evaporated, about 3 minutes. Transfer mixture to a 6-quart slow-cooker. Stir in tomatoes, chicken broth, and ground cumin. Add lamb shanks, nestling into tomato mixture. Cover and cook on LOW until lamb is very tender, about 6 hours.

Transfer lamb shanks to a plate. Pour cooking mixture through a wire-mesh strainer into a bowl; discard solids. Reserve strained cooking liquid. Set aside lamb shanks, covered.

Bring 2 cups reserved strained cooking liquid to a simmer in a small saucepan over medium. Whisk together cornstarch and 1 tablespoon water. Whisk cornstarch mixture into cooking liquid; cook until thickened and reduced by half, about 12 minutes. Set sauce aside; keep warm.

Bring 4 cups water to a boil in a medium saucepan. Whisk grits into boiling water; cook, whisking constantly, until smooth, about 45 seconds. Return to a boil; cover and reduce heat to medium-low. Cook until tender, about 20 minutes. Whisk in Parmigiano-Reggiano cheese, 1/4 cup chopped fresh chives, unsalted butter, and 2 1/2 teaspoons kosher salt.

Divide grits among 4 shallow bowls. Top each with a lamb shank; spoon sauce over top, and garnish with 2 tablespoons chopped fresh chives.