For the tomatoes: Preheat the oven to 375 degrees F. Line 2 sheet pans with parchment paper.
In a large flat bowl or pie plate, whisk together the flour, 1/2 teaspoon of the salt and 1/2 teaspoon of the Cajun seasoning. Put the breadcrumbs into a second pie plate and season with the remaining 1 teaspoon salt and 1 teaspoon Cajun seasoning. Pour the buttermilk into a third pie plate. Pat the tomato slices dry with a paper towel. Dredge them first in the flour mixture, shaking off any excess, then in the buttermilk, and then finally in the panko, patting to coat completely. Place onto the lined sheet pans, spaced about 1 inch apart. Spray the tops lightly with nonstick spray.
Bake for about 15 minutes, then flip the tomatoes, spray the tops lightly with nonstick spray, rotate the pans and bake until golden brown, an additional 15 minutes.
For the Cajun-spiced aioli: Meanwhile, sprinkle the minced garlic with salt and use the back of a knife to press it into a paste. Add the garlic and egg yolk to a stainless steel bowl along with the lemon juice and Dijon mustard; whisk to combine. Combine the oils in a container with a spout that will allow for easy controlled pouring. While whisking constantly, slowly add the oil in a slow steady stream (see Cook's Note). Stir in the Cajun seasoning and add salt to taste.
Serve the tomatoes on a platter with a bowl of the Cajun-spiced aioli.
If the mixture is too thick, slowly add water 1 tablespoon at a time until the mixture is about the consistency of mayonnaise.