Rinse your mushroom caps, and pat dry. Rinse the bell peppers, cut off the tops, and cut into four slices each (removing the seeds). Preheat your grill to high, and then grill the mushrooms and peppers over direct heat, a few minutes per side. Preheat your oven to 200, then plate the mushrooms, topping them with the pepper slices on four oven-safe dishes. Keep them warm in the oven while you prepare the rest of the ingredients.
To poach your eggs, bring a pot of water to a gentle boil. Add the vinegar and allow it to resume boiling. Add the eggs, allowing them to gently boil for a little over three minutes.
While the eggs are being poached, prepare the sauce. In a small mixing bowl, combine the egg yolks, lemon juice, mustard powder, and salt. Using a glass measuring container, microwave the butter on high for about a minute, until it is very hot and bubbling. With an immersion blender (or electric beaters) slowly pour the hot butter into the egg yolk mixture, blending until it is creamy and thick.
Remove the plates from the oven, being careful of the hot dishes! Top each mushroom with 3 slices of salmon, and 1 egg. Pour the sauce over each, dividing it equally among the 4 plates. Top with the dill and fresh cracked pepper.