Scrub the mussels under cold water; discard any that don’t close when you tap them. Put olive oil in a large pot over medium heat. Peel garlic and slice thinly.
Add the garlic to the oil and cook, stirring occasionally, until it sizzles, just a minute or two.
Put mayonnaise in a small heatproof bowl. Halve the lemon and squeeze the juice into the mayonnaise through a strainer or your fingers to catch the seeds; whisk to combine. Get a large strainer handy; line it with cheesecloth if you have it.
Add the mussels to the pot, turn the heat to high, and cover the pot. Cook, shaking the pot occasionally, until they all (or nearly all) open, 5 to 10 minutes. (You can discard those that remain tightly closed or gently pry them open with a butter knife to see if they release easily and look and smell normal.) Transfer the mussels to a serving bowl.
Strain the liquid remaining in the pan through the strainer into the mayonnaise mixture; whisk until smooth. Taste and add salt and pepper. Drizzle the mayonnaise sauce over the mussels and serve.