Remove andouille sausage from casing and mix with ground beef. Form into patties, pressing a divot into the center of each patty. Season both sides of patties with Cajun seasoning.
Place a small heat-safe skillet and small heat-safe sauce pan on your grill over medium heat.
Add your bacon to your small skillet. Keep an eye on it. The bacon will burn easily.
Place on med-high grill to cook. Burgers will take 10-12 minutes to cook total, depending on the thickness of your patty and temperature of your grill.
In your small saucepan, mix water, vinegar, and next 7 ingredients. Bring to a boil.
Slice squash into thin rounds using mandolin. Thinly slice onion and red pepper. Place all three vegetables in glass bowl. Pour boiling vinegar mixture over. Cover with plastic wrap for 15 minutes.
Start your Aioli in a food processor or blender. Mix garlic and next 4 ingredients until well blended and smooth. While machine is running (lid on) slowly drizzle in ½ cup olive oil until well incorporated. Season to taste with salt and black pepper.
Flip your burgers after 5 minutes, they will easily move when they are ready to be flipped.
When burgers are done cooking. Top with Havarti Cheese.
Grill buns for 2-3 minutes.
Assemble burgers in this order. Aioli, arugula, burger, cheese, squash pickles,