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Slice each chicken breast horizontally into two cutlets, yielding four pieces.
In a shallow dish, whisk together all-purpose flour, cornstarch, paprika, garlic powder, onion powder, cayenne pepper (optional), salt, and black pepper.
In another shallow dish, whisk eggs with whole milk.
Dredge each chicken cutlet in the flour mixture, dip into the egg wash, then coat again in the flour mixture. Set aside.
Heat vegetable oil in a heavy pot or Dutch oven over medium-high until it reaches 350°F.
Fry two chicken cutlets at a time for 5–7 minutes per side, until golden brown and internal temperature reaches 165°F. Drain on a wire rack and repeat with remaining cutlets.
Peel and cut Russet potatoes into 1-inch chunks. Place in a pot, cover with cold water, add a pinch of salt, and bring to a boil. Cook 15–20 minutes until fork-tender, then drain.
Return potatoes to the pot, add unsalted butter and whole milk, and mash until smooth. Stir in salt and black pepper.
In a saucepan, melt unsalted butter over medium heat. Whisk in all-purpose flour and cook 1–2 minutes to form a roux.
Gradually whisk in chicken broth. Simmer 5–7 minutes, stirring, until gravy thickens. Season with salt and black pepper.
Lightly toast brioche buns if desired. Place a fried chicken cutlet on each bottom bun.
Top each cutlet with a dollop of mashed potatoes, then drizzle with gravy. Close sandwiches with top buns and serve immediately.