Trim scallions and cut into 3-inch segments. Slice each segment lengthwise into very thin strips. Place in a bowl of ice water and let sit while you cook. They'll curl into ribbons naturally.
Pasta
Boil spaghetti in well-salted water until 1 minute shy of al dente. Reserve 1/2 cup pasta water before draining.
Brown Butter
In a wide skillet over medium heat, melt butter. Continue cooking, swirling occasionally, until the butter turns amber and smells nutty, about 3-4 minutes. The milk solids at the bottom should be golden brown, not black.
Sauce
Add garlic to the brown butter. Reduce heat to medium-low and cook slowly, about 30 seconds. Add soy sauce, mirin, and chili flakes to the pan. It'll sizzle hard. Add 1/4 cup reserved pasta water and stir to combine.
Finish Pasta
Add spaghetti directly to the pan. Toss vigorously over medium heat until the sauce emulsifies and coats every strand, 1-2 minutes. Add parmesan and toss again. Add more pasta water if it tightens too fast. Finish with a knob of cold butter and toss until glossy.
Plate
Twirl spaghetti into a high mound. Drain scallion ribbons and shake off water. Pile them loosely over the noodles so the curls show.
Notes
Watch the brown butter closely. It goes from golden to burnt in seconds. Pull it off heat the moment it smells nutty.
The ice water is what makes the scallions curl. Don't skip it and give them at least 5 minutes to do their thing.
Reserved pasta water is the emulsifier here. Keep it close and use it aggressively if the sauce looks greasy or tight.