Prepare your potatoes: Cut the scrubbed potatoes into 1/4-inch thick fries. Rinse them under cold water and pat them thoroughly dry with paper towels. This is crucial for crispiness!
Form the burger patties: Divide the ground beef into 4-6 equal portions (about 3-4 oz each). Gently roll them into balls without overworking the meat. Season generously with salt and pepper just before cooking.
Preheat your Blackstone griddle: Heat your griddle over medium-high heat. Add the vegetable oil and spread it evenly. Add the dried potato fries in a single layer, ensuring they aren't overcrowded. Season them with seasoned salt.
Cook the fries: Cook the fries for about 10-15 minutes, flipping them occasionally with your griddle spatula until they are golden brown and crispy. Remove them from the griddle and set aside on a plate lined with paper towels.
Smash the burgers: Place the seasoned beef balls onto the hot griddle, leaving some space between them. Immediately use a sturdy spatula or burger press to firmly smash each ball into a thin patty (about 1/2 inch thick).
Season and cook: Season the smashed patties generously with salt, pepper, paprika, garlic powder, and a pinch of onion powder if desired. Cook for 2-3 minutes per side, until a deep brown crust forms and the burgers are cooked to your desired doneness. Don't overcook!
Add cheese: During the last minute of cooking, place a slice of cheddar cheese on each burger patty. Cover with a lid or a metal bowl to help the cheese melt.
Prepare the sauce: While the burgers cook, whisk together the ketchup, mayonnaise, mustard, sweet relish, and paprika in a small bowl to create a simple burger sauce.
Toast the buns: Lightly butter the cut sides of the brioche buns and toast them on the griddle until golden brown. Watch them closely as they toast quickly.
Assemble the burgers: Spread the burger sauce on the bottom and top halves of the toasted buns. Place a cheesy smash burger patty on the bottom bun, and top with the other half of the bun. Serve immediately with the crispy griddle fries.