In a large skillet, melt 3 tbsp. butter over medium. Add the onions and cook, stirring occasionally, until caramel in color and sweet, about 25 minutes. Season with a little salt and pepper.
In a medium bowl, mix the turkey, herbs, mustard and a drizzle of oil; season. Form into 6 patties (thinner in the centers for even cooking). Heat a cast-iron skillet or griddle over medium-high.
Preheat the oven to 425 degrees . Bake the fries according to package directions until evenly browned and crispy. Preheat the broiler.
In a small saucepan, melt 2 tbsp. butter over medium-high. Season with pepper; whisk in the flour. Add the stock and bring to a bubble. Add the Worcestershire and cook, stirring often, until the gravy is thickened, about 15 minutes.
Cook the burgers, turning once, until cooked through, 10 to 12 minutes.
Put the bun bottoms on a baking sheet; top with the patties and some fries and cheese. Broil until the cheese melts, 2 to 3 minutes. Drizzle with gravy; add the bun tops.