Combine oils in a glass measuring cup.
Put egg yolks, salt, lemon juice, mustard, vinegar, sugar, and cayenne in a blender or food processor. With motor running, gradually add oil in a steady stream, starting with a very slow drizzle. Once sauce begins to thicken, add oil more quickly.
Scrape into a bowl and fold in chopped ingredients. Taste the sauce and adjust with salt, lemon, vinegar, or cayenne; you want the flavor to be zippy.
Make ahead: Up to 1 day, chilled.