In a food processor or a blender, combine the egg, egg yolk, cilantro (if using), lemon juice, garlic, mustard, and jalapeños, and process until smooth. With the motor still running, add the oil in a thin stream until a thick emulsion is formed. Add the salt and pepper, and transfer the mayonnaise to a nonreactive container. Use immediately, or cover and refrigerate for up to 3 days.

Note: I suggest using caution when consuming raw eggs because of the slight risk of salmonella or other food-borne illness. To reduce this risk, I recommend that you use only fresh, properly refrigerated, clean, grade A or AA eggs with intact shells, and always avoid contact between the yolks or whites and the outside portion of the eggshell.