Whisk mayonnaise and Cajun seasoning together in a small bowl until smooth. Spread over both slices of toasted bread.
Place andouille sausage in a skillet, cut-side down. Cook over medium-high heat for 2 minutes. Flip and cook for 2 minutes more. Transfer andouille to one slice of bread, leaving a small space in between the halves as a “raft” to hold the egg in place.
Heat oil in the same skillet for 1 minute over medium heat. Crack egg into the skillet and let cook undisturbed for 3 minutes. Flip with a spatula and cook for 2 minutes more. Place fried egg on top of the andouille and cover with the second slice of bread.