Pat the shrimp and scallops dry with paper towels and season on both sides with salt and pepper. Heat the oil in a large sauté pan over high heat. Add the shrimp and scallops, working in batches if necessary to avoid crowding the pan, and sear, flipping once, until golden brown, 1 to 3 minutes per side depending on size. Transfer to a bowl and reserve. Set aside the pan (no need to wash it).

Bring a large pot of salted water to a boil over high heat. Add the pasta and cook according to the package directions until it is 3 minutes shy of al dente. Drain well, reserving 1½ cups of the pasta cooking water.

Return the sauté pan to low heat, add the butter, and let it melt, then add the garlic and capers. Cook, stirring occasionally, until the garlic is soft and fragrant but not brown, 3 to 4 minutes. Turn the heat up to medium-high and add the wine. Let simmer until reduced by half, 2 to 3 minutes. Add ¾ cup of the pasta water to the pan (it will bubble up a bit) and bring to a gentle boil, whisking to emulsify and scrape up any browned bits on the bottom of the pan, about 1 minute.

Add the pasta to the pan and cook, tossing occasionally, until it is al dente and the sauce reduces and becomes thick and glossy, about 3 minutes. Remove the pan from the heat and stir in the cheese until it's melted, thinning out the sauce with reserved pasta water if needed. Stir in the lemon juice and parsley, then add the seafood and any juices that have collected in the bowl. Adjust the seasoning to taste. Adjust the seasoning to taste and serve.