Preheat the oven to 425F.
Place the asparagus on a baking sheet, drizzle with olive oil and season with salt, pepper, garlic and basil.
Bring a medium pot of water to boil on the stove.
Roast the asparagus at 425F until they begin to wrinkle (about 10-12 minutes).
To poach the eggs, turn the boiling water down a notch and add vinegar to the water.
Begin to stir the water to create a whirlpool, then place one egg into the center of the pool using a ladle.
Let the egg swirl and cook until cooked to your desired doneness (about 4-5 minutes) and remove with a slotted spoon.
Repeat with the second egg.
Split the asparagus between two plates, top each with one egg, and squeeze the juice of half a lemon over each before serving.