Preheat oven to 400°F. Whisk together miso, honey, 5 tablespoons of the olive oil, 3 tablespoons of the lemon juice, 1/2 teaspoon of the salt, and 3/4 teaspoon of the pepper in a bowl.
Stir together 4 to 6 cups water (enough to cover artichokes) and remaining 1/4 cup lemon juice in a large bowl. Trim about 2 inches from top of each artichoke. Cut each artichoke in half vertically. Remove fuzzy thistle from bottom with a spoon; discard. Trim any leaves and dark green layer from base. Rub edges with lemon wedge, and submerge each artichoke in lemon-water mixture as you finish trimming. (This will keep them from turning brown.) Cut artichokes into wedges. Toss together artichoke wedges and remaining 3 tablespoons olive oil, 1 teaspoon salt, and 1/4 teaspoon pepper. Spread in a single layer on an aluminum foil-lined rimmed baking sheet. Roast in preheated oven until golden brown and tender, 20 to 25 minutes, stirring halfway through roasting. Cool 15 minutes on baking sheet.
Grate walnuts on a Microplane grater to equal 1 cup. Using a sharp knife, cut asparagus diagonally into very thin slices, and place in a large bowl. Add artichokes, spinach, miso dressing, dill, and half of grated walnuts to asparagus; toss to coat. Top with remaining grated walnuts.