Peel the radish and use the peeler to make thin radish slices. Better long.
In a different recipient mix the other ingredients starting with vegan mayonaise substitute and the mustard, then the oils, the vinegar and the soy sauce, and finally the sesame seeds. Sesame oil tastes really strong, so only a little is enough.
Add the dressing to the radish slices and mix a little. Put in the fridge for at least 30 minutes or longer before serving, so the radish will be a little softer.
If you use Japanese mustard (karashi) note that it is spicy. 1/4 teaspoon is more than enough. Maybe half is good.

Radish (better long or Japanese giant radish),
Whole or grounded sesame seeds
Mustard (Japanese "karashi")
Vegetable oil with light taste
Sesame oil
Rice vinegar
Soy sauce
Vegan mayonaise