Place the egg, egg yolk, lemon juice, mustard, 1 tablespoon of water, and the salt in the bowl of a food processor or blender and process on high speed for 20 seconds. With the motor running, slowly pour in 3/4 cup of the vegetable oil and process until the mixture begins to thicken. When the oil has been incorporated, add 1 more tablespoon of water. With the motor running, add the remaining vegetable oil and the olive oil in thin streams and process on high until all the oil is incorporated. Adjust the seasoning to taste with salt and cayenne, if desired.
Chill the mayonnaise and use as needed. (The mayonnaise will keep, stored in an airtight nonreactive container in the refrigerator, for up to 24 hours.)
Makes about 2 cups