Prepare an ice-water bath. Cover eggs by 1 inch of water in a small saucepan. Bring to a boil, remove from heat, and cover; let stand 12 minutes. Transfer eggs to ice bath until cool. Drain, peel, and coarsely chop.
Meanwhile, prepare a new ice bath. Working in batches, blanch haricots verts in a pot of generously salted boiling water until bright green and crisp-tender, about 2 minutes. Transfer to ice bath until cool; remove. Return water in pot to a boil and blanch snap peas, in batches, until bright green and crisp-tender, about 1 minute. Transfer to ice bath until cool; remove. Cut snap peas in half and haricots verts into thirds.
Whisk together anchovies, lemon juice, and sour cream in a bowl. Slowly add oil in a steady stream, whisking until combined. Season with salt and pepper.
Divide spinach among plates and top with vegetables, eggs, and herbs. Drizzle with dressing, season with salt and pepper, and serve immediately.