Bring a large pot of water to a boil and cook the frozen udon according to its package directions while you work on the sauce. Check for separated strands with a springy bite rather than a soft, collapsing texture.

Slice the green onion thinly, keep the white and green pieces separate, and stir the cornstarch into the warm water. The slurry should look uniformly cloudy, with no dry pockets of starch.

Melt the butter in a warmed skillet over medium to medium-low heat, then stir in the garlic and white onion pieces for 1–2 minutes. Look for softened onion and a rounded garlic aroma while keeping the garlic pale, not brown.

Work the miso into the buttery mixture with a flat whisk, using brisk strokes to break up the paste. Keep the heat gentle and continue until no distinct lumps of miso remain.

Switch off the burner and whisk the stirred cornstarch-water mixture into the skillet. The sauce should become uniform, with a glossy surface instead of separate butter puddles.

Adjust the consistency by adding water one tablespoon at a time if thick, or returning the skillet to medium-low heat to reduce if thin. A ready sauce flows around the pan while leaving a light coating on the whisk.

Taste the sauce and add a little salt only if desired. Look for a balanced buttery, savory flavor rather than a sharply salty finish.

Drain the udon, toss it energetically through the sauce, and transfer it to a bowl topped with sesame seeds and the reserved green onion. Each strand should carry an even, shiny coating, with little loose sauce left in the skillet.