Trim dark green top from leek; discard. Slice white and pale green parts ½" thick. Place in a large bowl, cover with water, and rinse, separating rings to remove dirt. Drain; pat dry.
Heat oil in a large pot over medium. Cook half of the leek, stirring occasionally, until softened, about 2 minutes. Add garlic and ¼ cup curry paste. Cook, stirring occasionally to coat leek, until fragrant and curry is brick red, about 2 minutes. Add broth, tahini, honey, soy sauce, vinegar, and 2 cups water and stir to combine. Increase heat to high and bring liquid to a boil. Reduce heat to low, cover pot, and simmer 20 minutes. Taste soup; stir in more curry paste and season with salt.
Meanwhile, cook noodles in a large pot of boiling salted water according to package directions, stirring occasionally, until al dente. Drain and set aside.
Add kale, butter, and remaining leek to soup and simmer, uncovered, stirring occasionally, until kale and leek are softened, 5–7 minutes.
Divide reserved egg noodles among bowls; ladle soup over.
