For the smothered chicken: Melt the butter in a medium saucepan over medium-high heat. Add the Cajun seasoning and scallions and cook, stirring, 1 minute. Add the flour and cook, stirring, 1 minute more. Whisk in the chicken broth and milk and bring to a simmer, whisking often, then simmer until thickened, about 2 minutes. Add the shredded chicken and stir until hot. Season with salt and a generous amount of black pepper, then add a few shakes of hot sauce. Keep warm or reheat before serving.
For the waffles: Preheat a waffle iron on the hottest setting.
Whisk the flour, sugar, baking powder and salt together in a bowl. In another large bowl, whisk the sour cream, milk, butter and eggs together. Add the flour mixture to the wet ingredients and whisk until just combined.
Spray the waffle iron with cooking spray and make the waffles according to the manufacturer's instructions.
Cut or break the waffles in half (2 squares per sandwich) and fill evenly with the hot smothered chicken.