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1
Place the kombu squares in a saucepan, cover with fresh cold water, and bring to a gentle simmer over medium heat, uncovered, for 15 minutes — so the preliminary soaking and simmering coaxes the kombu into genuine pliability, which means the braising liquid will penetrate to the centre rather than sitting on a still-leathery surface. The kombu should have softened noticeably and lost its rigid, leathery resistance. Drain and return the kombu to the pan.
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2
Add the soy sauce, mirin, sake, and sugar, set over low-to-medium heat, and bring to a steady simmer, uncovered; stir only occasionally, since frequent disturbance will break the kombu pieces apart before they have absorbed the glaze. This is where the dish lives or dies: maintain a consistent, active simmer for 20–25 minutes, reducing until virtually all liquid has been expelled and the kombu is coated in a thick, sticky, lacquer-like glaze, because a pan left insufficiently reduced will produce tsukudani that is watery, short-lived in the jar, and flat in flavour, while a pan left unattended at the end will scorch in moments. The correct endpoint is a pan that is nearly dry, with the kombu pieces visibly glossy and clinging together.
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3
Remove the pan from the heat, scatter the sesame seeds over the kombu, and fold them in gently so the residual heat toasts them lightly and the glaze, still just fluid enough to move, binds them to the surface rather than letting them fall loose. Allow the tsukudani to cool completely in the pan, during which time the glaze will set further and the kombu will take on a deep, lacquered appearance.
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4
Transfer to a clean jar or covered container and store refrigerated for up to 2 weeks. Serve in tablespoon-sized portions alongside plain cooked rice, as the concentrated sweet-salty-umami register is calibrated for this small measure and larger portions are overwhelming.