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In a small saucepan over high heat, bring the cider vinegar and dashi, if using, to a simmer (if you aren't using dashi, just simmer the vinegar). Cook, adjusting the heat to keep it simmering, not boiling, for 3 to 5 minutes, until reduced to about 1 tablespoon.
Transfer the vinegar-dashi concentrate to a small bowl. Form a ring with a damp dish towel to rest your bowl on—this will keep it stable. Add the egg yolk and mustard and whisk to combine.
Whisking constantly, very gradually drip in the oil down the side of the bowl into the yolk mixture. The mixture should emulsify and thicken.* You can drizzle a bit more quickly once the mixture is very thick.
When all the oil has been incorporated, mix in the sugar and salt. You can add a little dashi or water to thin the mayonnaise so it will easily flow from a squeeze bottle but still hold its shape. It will thicken slightly once refrigerated. Transfer the mayo to a squeeze bottle. Refrigerate and use within 1 week.
*If at any point the mixture breaks and separates instead of getting thick and creamy, don’t despair. Put a fresh egg yolk in a bowl and slowly whisk the broken mixture into it, as if it were the oil.