"While I believe a burger can be made out of ground anything, or whole mushrooms, or patties of peas and beans, I never thought of stacking shaved beef—until now. This is a riff on a Philly steak, an ode to a roast beef dinner, and a burger all in one."
Preheat the broiler. In a medium bowl, stir all the sauce ingredients until blended.
In a medium skillet, melt 1 tablespoon butter over medium-low to medium heat. Add the onion; season with salt. Cook, stirring often, until the onion is softened, about 10 minutes. Add ½ cup water. Cook, stirring occasionally, until the water evaporates, about 5 minutes. Remove the onion from the heat.
In a small bowl, microwave the remaining 2 tablespoons butter on high until melted, 30 to 40 seconds.
Toast the buns under the broiler until golden brown, 1 to 3 minutes. Brush the cut sides of the buns with melted butter.
Heat a large griddle or cast-iron skillet over medium-high to high. Add a drizzle of the oil. Working in batches, add the beef in a single layer; season with salt, pepper, and granulated garlic. Using tongs to turn the beef, cook until browned and crisp, 1 to 2 minutes per batch; add more oil between batches. Transfer the beef to a platter; tent with foil to keep warm.
Divide the onion among the bun bottoms. Top each with some pickles, a few slices of warm beef, some sauce, and the grated cheese. Repeat the layering 2 more times with the beef, sauce, and cheese. Top with some cress and more sauce. Set the bun tops in place.