Fry up the bacon until it is crispy and set it aside on a paper towel to absorb excess fat.
In a bowl, whisk together an egg, the milk and several healthy grinds of black pepper. Drop in your brioche slices and flip them over and over until they are totally soaked with the egg mixture. Fry them up in the bacon fat until they are golden brown.
Allow the bacon fat to cool to room temperature, then strain it into a measuring cup. Add enough olive oil to make a cup total. In a bowl, combine an egg yolk and the mustard.
Whisk them together until they are fully combined. Slowly start adding your bacon fat/oil mixture, whisking the entire time. Take it slow because if you add too much fat all at once, you will lose your emulsion. Keep adding the oil until it’s all combined and you have a thick mayonnaise. Add the lemon juice or vinegar and whisk it in quickly. Add salt to taste.
On a piece of the French toast, spread some of your bacon mayonnaise. Then add the bacon, tomatoes and lettuce. Top it off with the second piece of french toast. Throw some cheese in there if you want.