Heat oven to 400° F.
Combine the potatoes, 3 tablespoons of the oil, 3/4 teaspoon salt, and 1/4 teaspoon pepper in a large bowl. Transfer to a rimmed baking sheet in a single layer. Roast, stirring once, until golden brown, about 40 minutes.
When the potatoes have 15 minutes left to cook, heat 1 tablespoon of the oil in a large skillet over medium heat. Sprinkle the tuna with 1/2 teaspoon salt, the crushed pepper, and thyme. Cook, turning once, to desired doneness, 2 to 3 minutes per side for medium-rare. Put the tuna on individual plates.
Wipe out the skillet and heat the remaining oil over medium heat. Add the shallots and cook 2 minutes. Add the spinach, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Cook, stirring, until wilted.
Serve with the tuna, potatoes, and lemon wedges.