Place a large heavy-bottomed Dutch oven over medium heat. Add the olive oil and heat to shimmering, then add the onion, garlic, and a pinch of salt. Cook until aromatic and just starting to soften, about 2 minutes.
Add the mushrooms and rosemary and cook, stirring occasionally, until the mushrooms have given up most of their liquid and are beginning to brown, about 5 minutes.
Add the flour and cook for 1 minute, stirring constantly. Add the red wine and stock to deglaze the pan, scraping with a wooden spoon to loosen the browned bits on the bottom. Bring to a boil and then reduce the heat to medium-low to maintain a gentle simmer. Add the barley and cook, stirring occasionally, until it's tender, about 1 hour.
Stir in the oat milk. Taste and add salt and pepper if needed. Garnish with dill and serve.