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Melt butter in a large saucepan over medium heat. Add leeks; cook 4 minutes, stirring occasionally. Add stock, thyme, salt, and pepper; bring to a simmer. Add peas and asparagus; cover and cook 3 minutes. Place pea mixture and 3 1/2 tablespoons parsley in a blender. Remove center piece of lid (to allow steam to escape); secure lid on blender. Place a clean towel over opening in lid. Blend until smooth.
While soup simmers, add water to a skillet, filling two-thirds full; bring to a boil. Reduce heat; stir in vinegar. Break each egg into a custard cup. Gently pour eggs into pan; cook 3 minutes. Remove eggs with a slotted spoon; place on a paper towel-lined plate.
Divide soup among 4 bowls. Place 1 egg in each bowl; drizzle evenly with oil. Sprinkle with remaining 1 1/2 teaspoons parsley. Serve with toast.