Heat the broiler to high with the rack about 6 inches from the broiler.
On a sheet pan, lined with a silicone mat or aluminum foil, toss the potatoes with 2 tablespoons oil, season with salt and pepper and spread out in an even layer.
Broil until crispy and brown and just getting tender, about 10 minutes, turning with a spatula halfway through.
Meanwhile, rub the remaining 2 tablespoons oil on both sides of the fish and season generously with salt and pepper.
Remove the sheet pan from the oven, move the potatoes to the edges and nestle the fish between them, skin-side up.
Broil until the skin is crispy, about 5 minutes.
Cut asparagus into the 1 ½-inch pieces, drizzle with olive oil, sprinkle with salt and pepper.
With a large spatula, flip the fish, add asparagus to pan, and continue to broil until the fish flakes easily but the center is still pink and glossy, around 5 minutes.
If the potatoes are done after cooking the fish on the first side, pull them from the oven and transfer to a large serving bowl. If not, leave them until the fish is cooked to your desired doneness.
While the fish cooks, combine the parsley, mixed herbs, anchovy, garlic, lemon zest, yogurt and mayonnaise in a blender and purée until thick and uniformly green.
Taste and season with salt and pepper as needed.
Remove the fish and potatoes from the oven.
Scrape the potatoes into a large bowl and toss with the asparagus and cucumbers; squeeze half the lemon over the top, season with salt and pepper and toss again.
Cut the remaining lemon half into 4 wedges.
Transfer the fish and vegetables to a platter and serve the dressing on the side; or divide the fish and vegetables among four plates and dollop with dressing.
Sprinkle with dill and serve with lemon wedge.
