In a shallow bowl or baking dish, whisk together the eggs, whole milk, melted butter, brown sugar, vanilla extract, cinnamon, and kosher salt until smooth and fully combined.
Heat a large skillet or griddle over medium-low heat. Add a small amount of butter and spread it over the cooking surface.
Dip each slice of brioche into the custard mixture, allowing it to soak for about 30–60 seconds per side. Brioche absorbs the custard quickly, so don't oversaturate it.
Add the turbinado sugar to a plate or shallow dish. Lightly coat both sides of each soaked brioche slice with the sugar, shaking off any excess. You want a thin, even coating rather than a thick layer.
Place the sugar-coated brioche onto the buttered griddle. Cook for 3–5 minutes per side, or until deeply golden and the sugar has caramelized into a crisp crust. Keep the heat at medium-low and adjust as needed to prevent the sugar from burning.
Flip carefully! Resist the urge to flip the French toast repeatedly. Let it cook undisturbed so the sugar has time to caramelize and form a crisp crust.
Continue with the remaining slices, adding more butter and turbinado sugar to the pan and plate as needed. Scrape caramelized sugar from your utensils and griddle between batches to prevent it from burning.
Place the finished French toast on parchment paper rather than directly onto a plate, as the hot caramelized sugar can stick as it cools. Use caution when handling, the caramelized sugar is extremely hot!
Serve warm with homemade whipped cream, fresh berries, maple syrup, powdered sugar, sprinkles, or your favorite toppings.