Cook wild rice according to package directions.

Heat butter and oil in a large pot over medium. Add mushrooms, leeks, and kale; cook until liquid released from mushrooms evaporates, 5 minutes. Stir in garlic; cook 1 minute.

Stir in flour; cook 3 minutes, stirring constantly. Slowly whisk in broth and half-and-half until no lumps remain. Bring soup to a simmer and cook until thick, 5 minutes; season with salt and pepper. Serve soup with rice and garnish with chives.