Wash and dry the fresh produce. Grate the gouda on the large side of a box grater. Thinly slice the radishes into rounds. Halve, peel, and thinly slice the shallot(s).

Place the beef in a bowl; season with salt and pepper. Gently mix to incorporate. Form the mixture into 4 flat patties (or 8 patties for 4 servings), about 1/4-inch-thick. Evenly top the center of 2 patties (or 4 patties for 4 servings) with the grated gouda, then cover each with the remaining patties. Firmly pinch the edges to completely seal the meat around the cheese; cup the sides and gently squeeze to shape into ½-inch-thick patties.

In a medium nonstick pan (or large for 4 servings), heat a drizzle of olive oil on medium-high until hot. Add the stuffed patties. Cook 4 to 5 minutes, or until browned. Flip the patties (carefully, as the oil may splatter). Cook 4 to 5 minutes, or until the patties are browned and cooked through. Leaving any browned bits (or fond) in the pan, transfer to a work surface. Carefully drain off and discard any excess oil.

Meanwhile, in a large bowl, combine the arugula, sliced radishes, crispy onions, parmesan, and dressing. Toss to combine. Taste, then season with salt and pepper if desired.

Working in batches if necessary, add the buns, cut side down, to the pan of reserved fond. Toast on medium-high 30 seconds to 1 minute, or until lightly browned. Transfer to a work surface. Assemble the burgers using the toasted buns, dijonnaise, cooked patties, pickles, and as much of the sliced shallot(s) as you'd like. Serve the burgers with the salad on the side. Enjoy!