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1
Combine the dashi, soy sauce, and mirin in a donabe (ceramic nabe pot) or heavy-bottomed vessel and bring to a steady simmer over medium heat. The broth should smell clean and savoury with a faint sweetness from the mirin — not raw or flat.
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2
Add the white stalk sections of the Tokyo negi to the simmering broth. Simmer gently for 5 to 7 minutes. The negi is ready when the white sections have turned translucent and yielding at the edge while retaining their shape, and the broth has taken on a perceptible sweetness from the allium. Do not rush this stage: simultaneously combining negi and tuna produces under-cooked negi alongside over-cooked tuna.
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3
Add the tuna pieces to the already-simmering broth. This is where the dish lives or dies: tuna protein coagulates rapidly at 60 °C, and the pieces must remain in the broth for no more than 2 to 3 minutes. Remove the pot from direct heat the moment the exterior of each piece has turned opaque and pale while the centre remains visibly pink — medium-rare. Well-cooked tuna produces a rubbery, dry texture entirely contrary to the dish's intention.
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4
Bring the donabe to the table immediately. The retained heat of the ceramic vessel continues gentle cooking during communal eating; this is the intended serving vessel. Lay the reserved negi green tops across the surface as garnish. The broth should smell of ocean richness tempered by sweet allium.
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5
When the tuna and negi are consumed, add udon noodles or rice to the residual broth for the shime course. The tuna-negi-dashi broth is the most flavour-concentrated element of the meal; the shime is complete when the noodles or rice have absorbed the broth and the liquid is reduced to a glossy, intensely savoury coating. Taste for seasoning before serving.