First, place a saucepan of water on a high heat and add a good pinch of salt. Bring up to the boil then pop in the new potatoes, cover and boil for about 20-30 minutes until the potatoes are tender. While the potatoes are cooking, make the dressing by mixing the olive oil, the lemon juice, the chopped marjoram or tarragon, whichever you’re using, and some salt and pepper to season. When the potatoes are cooked, cut them into quarters while hot, place in a bowl and drizzle with some of the lemon and olive oil dressing and season with salt and pepper. Set aside but do not chill.
Place more water, with another pinch of salt, in the saucepan and bring to the boil on a high heat for cooking the asparagus. Prepare the asparagus by snapping off the woody ends at the base of the stalks, discard the snapped end. Next, using a peeler, peel away the slightly tough outer peel from about 2-3cm at the base of the stalks. When the water comes to the boil, drop in the asparagus and boil, uncovered, for about 4-6 minutes until tender. Remove the asparagus from the water but keep the water in the pot (for cooking the eggs) and place the asparagus stalks on a plate and immediately drizzle with some of the dressing.
Bring the water back to a boil again and crack in the eggs, turn the heat to medium to let the water simmer and poach the eggs for a few minutes until the whites are set and the yolks still soft. While the eggs are cooking, place the lettuce leaves in a bowl and drizzle with nearly all of the remaining dressing – enough to just make the leaves glisten. Divide the leaves between the plates and arrange the asparagus over or around or slightly peeping out from under the lettuce, then arrange the potatoes over or around and then place a perfectly poached egg sitting on top of each salad. To finish, peel a few shavings of the Coolea or Parmesan cheese, whichever you are using, for each plate and arrange over the top, drizzle with the last of the dressing, and serve.