Bring a large pot of heavily salted water to a boil over high heat. Add 1 pound dried spaghetti and cook according to package directions until just al dente. Reserve 2 cups of the cooking water, then drain the pasta.
Place 3 ounces finely grated Pecorino Romano cheese in a small bowl. Drizzle in ¼ cup of the pasta water and mash with a fork until the cheese becomes a soft paste.
Heat 2 tablespoons olive oil in a large skillet over medium heat until shimmering. Add 1 teaspoon coarsely crushed black peppercorns and cook until fragrant, about 30 seconds. Stir in 3 to 4 tablespoons gochujang and ¾ cup of the pasta water, and bring to a simmer.
Add the spaghetti and toss until coated. Turn off the heat. Add the cheese paste and use tongs to vigorously toss the pasta, adding more pasta water a few tablespoons at a time as needed to create a creamy sauce that clings to the pasta. Serve immediately with more grated Pecorino if desired.
