Reduce heat to medium. Add shallots, chile, and oregano, and sautΓ© until shallots are sizzling, about 30 seconds. Add potatoes and garlic, cover, and cook, shaking pan often, until edges of potatoes begin to brown, about 5 minutes. Uncover, add wine, and cook, stirring occasionally, until liquid is nearly evaporated, 5 to 6 minutes.
Step 4
Add tomatoes, cover, and cook until tomatoes start to split and release their juices and potatoes are tender, about 8 minutes. Add shrimp, and cook, turning occasionally, until shrimp are cooked through and tomatoes are soft, 2 to 3 minutes. Season with salt and white pepper; stir in basil. Divide among 4 shallow bowls, drizzle with remaining 2 tablespoons oil, and serve.
