Heat oil in a Dutch oven. Add mushrooms, celery, carrots, and leeks. Cook, stirring often, until softened about 7 minutes.
Stir in coconut milk, salt, and pepper and cook for 2 minutes. Add white wine and thyme and simmer 2 minutes more.
Add broth to Dutch oven and bring to a boil. Add instant rice and peas. Reduce heat to a simmer. Cover and cook until rice is tender, about 7 minutes, then stir in dill. Garnish with pea shoots.