Bring 4–6 cups of salted water to a rolling boil in a large pot. Add the spaghetti and cook for 9–11 minutes, until al dente. Drain and set aside.
In a small bowl, whisk together the soy sauce, brown sugar, mirin, grated ginger, and Thai roasted red chili paste until smooth and well combined.
In a medium skillet over medium heat, warm the sesame oil. Add the minced garlic and sauté for about 30 seconds, just until fragrant. Be careful not to burn it.
Pour the prepared sauce into the skillet with the garlic. Stir and cook for 30 seconds to 1 minute, until the sauce begins to bubble slightly and thicken enough to coat the back of a spoon.
Add the drained spaghetti to the skillet and toss thoroughly until all the noodles are evenly coated in the sauce. Remove from heat.
Season the shrimp as desired. In a separate pan over medium-high heat, cook the shrimp for about 2 minutes per side, or until pink and opaque.
Top the noodles with the cooked shrimp. Sprinkle with sliced green onions and sesame seeds before serving.