Cook the spaghetti according to the package directions; drain and return it to the pot.
Meanwhile, heat the oil in a large pot over medium heat. Add the garlic and ginger and cook until fragrant, 1 to 2 minutes.
Add the broth and chicken and bring to a boil. Reduce heat and simmer, stirring occasionally, until the chicken is cooked through, 20 to 25 minutes. Shred the chicken, reserving the broth.
Add the chicken, broth, kimchi, and ½ teaspoon salt to the pasta and stir to combine.
Serve topped with the snow peas and fried eggs.