1. Preheat oven to 425 degrees. On a rimmed baking sheet, toss potatoes, onion, carrots, beet, and rosemary with 2 tablespoons oil; season with salt and pepper. Roast vegetables until golden and tender, about 30 minutes.

2. Meanwhile, in a straight-sided skillet, heat 2 inches water over medium until a few bubbles rise to the top. Crack eggs, one at a time, into a small bowl and gently pour into skillet. Cook, undisturbed, until whites are just set and yolks are still runny, 3 to 4 minutes. With a slotted spoon, transfer eggs to a paper-towel-lined plate.

3. Transfer roasted vegetables to a large bowl and toss with lettuce, lemon juice, and remaining tablespoon oil; season with salt and pepper. Divide among four plates, top with poached eggs, and serve with lemon wedges.