Step 1: Heat oil in a large pot over medium-high heat. Add the smoked sausage and/or andouille and saute' to render the fat.
Step 2: Add the garlic, onion, celery, and thyme and saute' until celery wilts slightly.
Step 3: Deglaze the pan with the white wine, stirring up the brown bits in the pot with a wooden spoon.
Step 4: Add the corn, potatoes, crawfish stock (optional), half and half (if using), and milk (minus 1/2 cup for thickening).
Step 5: Stir the mixture, taste and season to your liking with salt, pepper, Creole seasoning and hot sauce.
Step 6: In a small bowl, whisk the flour with the reserved cup of milk to create a paste, then add slowly to soup while whisking.
Step 7: Add the crawfish and simmer for about 10 minutes. Stir in green onions. Taste and adjust seasonings.