Combine water with dried anchovies, onion, garlic, shiitake mushrooms, and kelp in a medium pot. Bring to a boil over high heat. Lower the heat to medium-low and let simmer for 20 minutes. Strain the stock, discarding the aromatics. Set aside.
Bring a pot of water to a boil over high heat. Add the dumplings and cook until the dumplings and float to the top, about 8 minutes. Drain and set aside.
Heat 1 tablespoon oil in a medium saucepot over medium-high heat. Add the garlic, and saute for 30 seconds or until aromatic. Add kimchi, anchovy stock, and chili flakes, and bring to a boil. For a mild stew, add 1 tablespoon chili flakes. For extra hot, add 4 tablespoons or more.
Add the tofu, breaking up the large pieces with a wooden spoon. Season the stew with fish sauce, soy sauce, and black pepper. Stir to combine and let simmer for 5 minutes. Add the dumplings and crack the eggs directly into the pot. Simmer for another minute then remove from heat. Garnish with green onions and serve with rice.