Bring a pan of salted water to the boil, then reduce to a simmer. Heat a griddle pan over high heat and toast the bread. Move each slice to a plate or board and drizzle with a little olive oil. Griddle the asparagus for a few minutes each side, sprinkle with the paprika and divide between the pieces of toast.
Meanwhile, poach the eggs in the simmering water for 3 minutes. Using a slotted spoon, remove the eggs and drain, then place them on top of the asparagus. Grate over the parmesan, and sprinkle with mint leaves.