Mash garlic into a paste. Whisk together yolk, lemon juice, and mustard in a bowl. Combine oils and add, a few drops at a time, to yolk mixture, whisking constantly, until all oil is incorporated and mixture is emulsified. Whisk in garlic paste and season with salt and pepper, to taste. If the aioli is too thick, whisk in 1 or 2 drops of water.;
Warm lobster in melted butter and set aside. Toast white bread. Smear with aioli. Layer bacon, lobster, tomatoes, sprouts and avocado.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.