In a large saucepan, heat oil over medium. Add barley and leeks; cook until beginning to soften, 5 to 7 minutes. Add wine; cook until evaporated, about 5 minutes. Add 2 cups water; bring to a boil and season with salt and pepper. Reduce heat; simmer until liquid absorbs, about 10 minutes.
Add broth and continue to cook, stirring occasionally, until barley is tender and creamy, about 10 minutes. Add asparagus; cook until tender, about 5 minutes. Stir in peas until heated through. Add Parmesan and mint; season with salt and pepper. Garnish with mint leaves.