Bring a large pot of lightly salted water to a rolling boil. Cook the spaghetti at a boil until tender yet firm to the bite, about 12 minutes; drain and set aside.
Heat the vegetable oil in a large skillet over medium heat.
Cook and stir the garlic and ginger in the hot oil until fragrant, about 1 minute.
Stir the kale into the garlic and ginger mixture; continue cooking and stirring 1 minute more.
Add the drained spaghetti to the kale mixture; toss while cooking about 30 seconds before adding the soy sauce and water. Bring the mixture to a simmer and cook for 2 minutes.
Make a well in the middle of the spaghetti mixture. Pour the eggs into the well; gently fold the eggs into the noodle mixture as they cook until they are evenly distributed through the noodles and cooked, about 5 minutes.
Season with the crushed red pepper flakes. Drizzle the sesame oil over the noodles to serve.