Stir together water, rice, and 1/4 teaspoon salt in a saucepan. Bring to a boil. Reduce heat and simmer, covered, about 15 minutes. Remove from heat. Let stand, covered, about 10 minutes. Let cool, about 30 minutes. (You should have 3 cups rice.)
Meanwhile, line a tray with wax paper or parchment paper. Put rice, kimchi, flour, egg, and remaining salt in a large bowl. Using your hands, gently combine. Shape rice mixture into 8 patties, 1/3 cup mixture each. Put patties on the prepared tray.
Heat 1 tablespoon oil in an extra-large nonstick skillet over medium-high heat. Cook 4 patties, turning once halfway through, until brown and crispy, about 8 minutes. Repeat with remaining oil and 4 patties.
Meanwhile, steam edamame according to package directions. Drain and transfer to a bowl. Add chili oil and soy sauce and toss until coated.
For sauce, stir together mayonnaise, gochujang, and vinegar until smooth.
Serve patties with steamed edamame and sauce. Top with green onion (if using).