Whisk the eggs and milk in a medium bowl. Sprinkle in the Bisquick and whisk until smooth. Let the batter rest 10 minutes to hydrate while you prepare the filling.
Set a rack in the middle of the oven and preheat to 385°F. Lightly coat a 9-inch deep pie dish with nonstick spray.
Warm a large skillet over medium heat. Add the ground beef and onion and cook, breaking up the meat, until the beef is mostly browned and the onion is tender, 6–8 minutes. Stir in the burger seasoning, salt, and pepper. Add the garlic, mustard, Worcestershire, and smoked paprika; cook 30–60 seconds more until fragrant. Drain excess fat and let the mixture cool for 2 minutes off the heat.
Give the batter a quick whisk. Pour about one-third of it into the prepared pie dish and tilt to coat the bottom in a thin layer. Par-bake this base for 5 minutes to set.
Spread the beef mixture evenly over the set base. Sprinkle half of the cheddar over the beef. Pour the remaining batter on top, tilting the dish so it flows into the gaps without disturbing the layers.
Bake until the pie is puffed around the edges and the center just sets, 18–22 minutes. Scatter the remaining cheddar over the top and return to the oven until melted and lightly golden, 3–4 minutes (or broil 1–2 minutes for extra color).
Rest the pie 10 minutes before slicing. Top with chopped dill pickles and a drizzle of ketchup, if you like.
Tip: For the cleanest slices, blot any excess fat from the skillet before filling the pie dish and wipe the knife between cuts.