In a skillet, heat 1/4 inch of vegetable oil until shimmering. Add the capers and fry over moderately high heat, stirring, until browned and crisp, 3 minutes. Using a slotted spoon, transfer the capers to paper towels to drain.

In a saucepan of salted boiling water, cook the spaghetti until al dente. Drain, reserving 1 cup of the cooking water.

Meanwhile, in a medium bowl, whisk 2/3 cup of the olive oil with the grated cheese, lemon zest and lemon juice.

In a very large skillet, heat the remaining 1 tablespoon of olive oil until shimmering. Season the shrimp with salt and pepper, add to the skillet and cook over moderately high heat, turning once, until just opaque, 4 minutes. Reduce the heat to moderate and add the pasta, lemon sauce, half of the reserved pasta water and the 1/2 cup of basil. Cook, tossing, until the pasta and shrimp are coated, 2 minutes, adding more of the pasta water if the spaghetti is dry; transfer to shallow bowls, garnish with basil and the fried capers and serve.