Place the garlic, mustard, and egg in the bowl of a food processor fitted with a blade attachment. Process until evenly combined, about 10 seconds.
With the motor running, slowly add the olive oil in a thin stream, followed by the grapeseed or vegetable oil, until completely combined, about 2 minutes. Stop the processor, add the lemon juice, season with salt, and pulse until thoroughly mixed. Stop and scrape down the sides of the bowl with a rubber spatula, then pulse until all ingredients are evenly incorporated. Let sit for at least 30 minutes before using. Refrigerate in a container with a tight fitting lid for up to 3 days.
Grilled Cheese Directions
Lay slices of bread on a board and spread each one lightly with butter. Flip the slices and spread each one generously with the aioli mixture.
Grate all the cheeses in a food processor fitted with the largest grating disk and combine. Pile 1/3 of grated cheese evenly on one slice and top with the (bacon, tomato and apple) and top that with the remaining cheeses and bread slice. So the buttered side is exposed.
When the flat top is really hot, put a bit more butter on the cooktop and cook the sandwiches for 3 to 5 minutes until the bread is toasted and the cheese is melted flip to allow bread to toast on other side. Allow cooling for 2 minutes. Cut in half and serve warm.