Bring a large pot of heavily salted water to a boil. Meanwhile, thinly slice scallions; set aside about 1 tablespoon for garnish. Finely chop garlic cloves. Cut the chicken into uniform, bite-size pieces.

Place gochujang, soy sauce, and honey in a small bowl and stir to combine. Place heavy cream and egg yolks in a liquid measuring cup or medium bowl and whisk to combine. Finely grate Parmesan cheese and set aside.

Add dried fettuccine to the boiling water and cook according to package directions until al dente, 8 to 10 minutes. Meanwhile, heat olive oil in a large skillet over medium-high heat until shimmering. Add the chicken and spread into an even layer. Season with kosher salt and black pepper, and cook undisturbed until the bottom is golden-brown, 5 to 6 minutes. Add the scallions, except the garnish, and garlic and sauté until fragrant, 1 to 1 1/2 minutes. Turn off the heat.

When the pasta is ready, reserve 1/2 cup of the pasta water, then drain the pasta. Return the skillet to medium-high heat. Add the gochujang mixture and cook, stirring frequently, until thickened and bubbling, about 1 minute. Add the cream mixture and stir to combine. Add the Parmesan and simmer until the sauce thickens and the color of the sauce turns glossy and a rosy-pink color, 3 to 5 minutes.

Add the pasta and toss to coat, adding some of the reserved pasta water a little at a time if needed to get a saucy consistency. Garnish with the reserved scallions and more Parmesan if desired.