Bring a large pot of water, seasoned with 1/2 tsp. salt, to a boil. Meanwhile, fill a sink or bowl with cold water.
Boil mustard greens until bright green, about 2 minutes. Drain well in colander and press to remove as much liquid as possible. Set aside.
Heat garlic and oil together in a medium frying pan over medium heat until garlic is golden and fragrant, 5 to 7 minutes. Use a slotted spoon to transfer the garlic chips to a plate.
Raise heat to high and quickly add sugar, remaining 1/2 tsp. salt, and the vinegar. Cook until sugar dissolves, about 3 minutes. Add reserved mustard greens and stir to coat. Cook until most of the liquid is gone, about 3 minutes. Serve greens over grits and sprinkle with garlic chips.