Insert the metal blade in your food processor. Process the egg, egg yolk, lemon juice, wine vinegar, water, sugar and flour until smooth, about 20 seconds. Transfer the egg mixture to a small (7 or 9 inch ) nonstick skillet, and heat over very low heat while gently begins to thicken and resemble a custard sauce, remove from the heat while still stirring and set in a pan of ice and water to stop the cooking. Stir until the egg mixture is cooled; let rest for 5 minutes. Wash the work bowl, metal blade and lid in hot soapy water and dry thoroughly.
Insert the metal blade. Process the cooled egg mixture with the dry mustard, salt and cayenne until blended, 1 minute. Scrape the work bowl. With the machine running, add the olive oil through the hole in the small feed tube very slowly. Scrape the work bowl. With the machine running, add the vegetable oil through the hole slowly into the emulsion. The mayonnaise will thicken as the oil is added. Taste and adjust seasonings as necessary. Remove to a container, cover and keep refrigerated. Keeps 3-4 days refrigerated.