Loosen the udon according to the package directions. Drain thoroughly. If using vacuum-packed noodles, warm them only until the strands separate; do not cook them until soft.
Stir the gochujang, soy sauce, brown sugar, reserved kimchi liquid, and 2 tablespoons water together in a small bowl.
Heat a wide skillet over medium-high heat. Add the neutral oil, kimchi, scallion whites, and garlic. Cook for 3 to 4 minutes, stirring often, until excess liquid has evaporated and the kimchi no longer leaves a watery pool in the pan.
Add the drained udon and the sauce. Toss with tongs for 2 to 3 minutes, lifting rather than pressing the noodles, until the sauce coats the strands. Add 1 tablespoon water at a time only if the sauce tightens before the noodles are hot.
Remove the skillet from the heat. Fold in the toasted sesame oil. Divide between two bowls and finish with scallion greens and sesame seeds.