Use a grater or microplane to grate the garlic. Add the garlic to a small or medium bowl, along with the salt, lemon zest, lemon juice, and mustard. Allow to rest for 10 to 15 minutes to mellow out the pungency of the garlic.
Drain the tofu, measure it out, and gently pat it dry several times.
Add the aquafaba to a food processor. Blend until the mixture is frothy (if your machine is large, you may need to hold the base of your food processor and slightly tilt the machine to the side so you can froth the aquafaba). Add the garlic-lemon mixture and blend again. Add the tofu and blend once more until the mixture is smooth.
With the center cap or spout open, slowly drizzle in the oil (do not add the oil all at once). Blend until very smooth.
Taste, adding more salt or lemon juice as desired. Add in the white pepper or black pepper. Transfer to a jar, cover, and refrigerate for a few hours to thicken and to allow the flavors to meld together (you can serve it immediately though). Stays good in the fridge for 5 to 6 days.