Whisk the yolks, lemon juice, and mustard together until frothy. Slowly whisk in the oil, a drop at a time, until the sauce begins to emulsify. Add the remaining oil in a slow, steady stream, whisking constantly. Season with salt and pepper to taste.
Aïoli: Smash and mince 3 to 4 cloves garlic. Make the basic mayonnaise, adding the garlic to the yolks and lemon juice, omitting the mustard, and using olive oil.
Lemon-herb: Add 1 tsp. grated lemon zest and 1/4 cup chopped fresh herbs (such as dill, parsley, tarragon, chervil, or basil) to the basic recipe.
Rouille: Roast, peel, seed, and purée 1 medium red bell pepper. Whisk together with 1/2 tsp. tomato paste and 1 large clove garlic, smashed and minced. Make the basic mayonnaise using olive oil and omitting the mustard. Whisk in the bell pepper and tomato mixture and season with 1/4 tsp. hot sauce, or more to taste.
Curry: Heat 1 Tbs. curry powder in 1 Tbs. vegetable oil for 1 min. and add to the basic recipe. Other spice blends, such as chili powder or garam masala, also work well.
Rémoulade-style: Make the basic recipe, increasing the mustard to 2 Tbs. and adding 2 Tbs. white-wine vinegar, 2 Tbs. tomato paste, 2 minced cloves garlic, 2 tsp. sweet paprika, and a dash hot sauce to the egg yolk and lemon juice base. Use all vegetable oil and finish by stirring in 1/4 cup each of finely chopped scallions, capers, and parsley.
Ginger-lime-cilantro: Make the basic recipe, substituting 1Tbs. lime juice for the 1 tsp. lemon juice, omitting the mustard, and using 1-1/4 cups peanut oil with 1/2 tsp. toasted sesame oil. Grate a 3- to 4-inch piece of peeled ginger and squeeze out about 2 Tbs. ginger juice (or use bottled). Whisk in the juice and 3 Tbs. chopped fresh cilantro.
Chipotle chile: Stir 1 Tbs. minced canned chipotle chile and 1 minced small clove garlic into the basic recipe. Season with black pepper in place of white. If desired, stir in 3 Tbs. chopped scallions or cilantro.