Cut chicken breasts in half crosswise. In a very large dutch oven over medium heat, fry chicken in vegetable oil in batches, turning often, until golden brown on all sides. Transfer to paper towels to drain.
In the same pan with the oil and chicken fat, add onions, celery, red bell pepper, jalapeño pepper, garlic, and bay leaves. Cook, stirring often, until onions begin to soften and turn translucent.
Add paprika, chili powder, cayenne pepper, and thyme. Stir to combine and cook until spices become fragrant and onions begin to brown.
Add the sliced alligator sausage, and cook until it just begins to brown.
Add uncooked rice and stir, cooking a few minutes more until rice begins to toast.
Add tomatoes, chicken stock, and shrimp. Stir well to combine, then add reserved chicken pieces, and stir well again.
Bring to a simmer, then cover, reduce heat, and cook until all liquid is absorbed, about 20-30 minutes.
Remove from heat, and stir in chopped parsley. Serve topped with sliced scallions.