Heat olive oil in a large Dutch oven over medium heat. Add leeks and carrot. Cook 7-8 minutes, or until softened and beginning to turn golden brown. Stir in mushrooms. Cook until juices are released, 3-5 minutes. Add wine, thyme, and sage. Cook 2-3 minutes more, or until liquid has almost evaporated.
Add mushroom stock and groats. Bring to a boil, then reduce heat to low. Partially cover and cook 15 minutes, or until groats are tender.