How to Make It
Using a knife, make a 1/4-inch-deep cut along the back of each shrimp shell from head to tail. Leave shell and head intact. Using a small moistened paper towel and a paring knife, remove and discard the vein. Pat shrimp dry, and set aside.
Heat 1/4 cup oil in a very large skillet over medium-high until shimmering and very hot. Add shrimp, and sear until shells are scorched, about 1 minute. Transfer shrimp to a plate.
Reduce heat to medium. Add shallots, chile, and oregano, and sauté 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.
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.