Place the egg, mustard and a pinch of salt into a food processor and blend together.
Mix the two oils together and add half the oil to the egg and mixture, very slowly, while still blending. The mixture will thicken.
With the blender still running, add the lemon juice and resume pouring in the remaining oil.
Add the vinegar and season, to taste, with salt and freshly ground black pepper.
To serve, place the radishes on a plate, with the salt and mayonnaise beside in separate ramekins.