Heat oven to 425° F. Combine the potatoes, 3 tablespoons of the oil, ¾ teaspoon salt, and ¼ teaspoon pepper in a large bowl.
Transfer to a rimmed baking sheet in a single layer. Roast, turning once, until golden brown, about 35 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 ½ 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 for 2 minutes.
Add the spinach, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, stirring, until the spinach is wilted. Serve with the tuna, potatoes, and lemon wedges.