Use a sharp knife to slice the inch or two off of the top of a whole head of garlic. This will expose the cloves within their paper shells. If a few cloves remain hidden, use the knife to cut into the tops of them.
Place a square of foil onto a baking sheet or in an aluminum pie plate. Put the head of garlic in the center of the foil. Drizzle 2 tablespoons of the olive oil over the top of the head, covering each clove.
Pull the edges of the foil up around the garlic to enclose it. Place the sheet or pie plate in the oven and roast the garlic for 45 minutes. The cloves should be browned and soft.
Remove the garlic from the oven and allow it to cool to room temperature. Once it is cool enough to handle, squeeze the cloves out of the papery skins. Discard the skins.
Hold the cloves aside, or refrigerate them in a sealed container until ready to make the aioli.
Make the aioli:
Place the remaining small, fresh clove of garlic, the roasted garlic, egg yolks, vinegar, mustard, salt, and pepper into the bowl of a food processor.
Blend the ingredients together until they are smooth.
With the processor running, slowly add in the remaining olive oil in a stream to blend it into the other ingredients. (If you can't add the oil with the machine running, add it a little at a time, blending between additions.)
The mixture should be a smooth and uniform emulsion. If it is too thick, add a little water and then blend again. If the emulsion breaks, stop adding oil and blend until the aioli is smooth again.
Assemble the sandwiches:
Slice the tomatoes into ½ inch thick slices. Use a paper towel to blot the slices dry.
Toast the slices of bread to your liking: for BLTs I like the bread toasted lightly, but go for darker if that's what you prefer.
Lay out 4 toast slices. Spread each slice with about 1 tablespoon of garlic aioli.
Place pieces of lettuce on each slice of bread. Top the lettuce with the sliced tomato, 2 slices per sandwich. Sprinkle a pinch of salt and pepper (to taste) over the tomatoes. Top the tomatoes with 4 slices of bacon on each sandwich. (The bacon slices can be cut in half to make them fit better if you'd like.)
If desired, spread more aioli on the remaining slices of toast. Top each sandwich with toast slices, aioli side down.
Use a sharp knife to slice each sandwich in half, and skewer each half with sandwich picks. Place the sandwiches on individual plates and serve them immediately.
This recipe makes about 1 cup total of garlic aioli. which can be kept in the fridge for 3-5 days.
Notes
Allow time for the garlic to roast. This step can be done in advance: store the roasted cloves in a sealed container in the fridge.
The garlic aioli can be made in advance as well, and stored in the fridge for 3-5 days.
This homemade aioli is made with raw egg yolks. Eating sauces and foods made with raw or under-cooked eggs is not recommended for pregnant women, elderly, children under the age of 4, or individuals with compromised immune systems.