Place racks in the upper third and center of the oven and preheat to 425°F.

Stand the peppers up on their bottom ends, then cut down and around the sides to remove the pepper "walls." Remove and discard the seeds, stems, and membranes. Place them in the center of a rimmed baking sheet.

Place the onion and garlic on a second rimmed baking sheet.

Drizzle pans with 2 tablespoons olive oil each, 1/2 teaspoon smoked paprika each, 1/4 teaspoon salt each, and 1/4 teaspoon pepper each. Toss both pans to coat, rubbing the oil and seasoning all over.

Arrange the peppers so they are lying flat, skin-side up. A little overlap is OK. Spread the onions and garlic into a single layer.

Roast the onions and garlic on the center rack for 20 to 25 minutes, stirring once, until the onions are soft and started to turn dark at the edges. Roast the peppers on the upper third rack for 40 to 45 minutes, until the skins are blistered and blackened in places (no need to flip the peppers).

Transfer the peppers to a bowl, cover tightly (I set a large plate or a pot lid on top), and let steam for 15 minutes to soften. Uncover and slip off the skins with your fingers. Don't worry if you don't get every last one.

Working in two batches, add half of the peppers, onion, and garlic to a food processor fitted with a steel blade or to a blender, along with half of the crushed tomatoes. Purée until completely smooth, then pour into a heavy-bottomed pot such as a Dutch oven. Repeat with the remaining vegetables and tomatoes.

Stir in the basil and remaining 1/4 teaspoon salt. Bring to a simmer over medium heat and cook, partially covered to catch any spatters, until the soup is heated through and smells super fragrant, stirring periodically.

Adjust salt and pepper to taste. Serve warm, topped with fresh basil and Greek yogurt.