Preheat oven to 450°F. Wash and quarter potatoes (eight wedges total); lightly mist with olive oil. Place in a shallow baking pan and sprinkle with salt and pepper. Roast for 20 to 30 minutes in the preheated oven, turning occasionally to brown on all sides.
While potatoes roast, trim, wash, and dry the asparagus. When potatoes are halfway done, add the asparagus to the baking pan, mist with olive oil, season with salt and pepper, and roll to coat. Roast about 10 minutes, or until barely tender when pierced with a knife. When potatoes and asparagus are done, remove from oven and cool. Cut the asparagus into 1 inch pieces.
While vegetable are roasting, place champagne vinegar, mustard, and olive oil in a small jar with a tight lid and shake to combine.
Divide mixed salad greens between two plates, top each with half of the asparagus and four potato wedges. Sprinkle each equally with salad dressing.
Bring 2 quarts of water with a pinch of salt and 1 teaspoon white vinegar to a simmer. Poach the eggs for 2 to 3 minutes; remove eggs with slotted spoon and place one on each salad plate; serve immediately.