Melt 2 tablespoons butter in a skillet over low heat; add onions. Simmer for 30-45 minutes to caramelize and develop a deep golden colour, adding small amounts of water as necessary to prevent burning. Add brandy, simmer, and stir until the alcohol has cooked off and the flavours, combine 3-5 minutes; set onions aside and keep warm.
In a medium mixing bowl, hand blend thyme and shallots with the ground beef, form into four patties, and season both sides with salt and pepper. Refrigerate 10-15 minutes.
Brush cut sides of buns with butter and pan sear or grill to golden brown.
Pan sear or grill patties for 4-5 minutes on one side, flip and top each with caramelized onions and Gruyère cheese. Continue to cook another 4-5 minutes until the burger reaches an internal temperature of 160°F (71°C).
Spread Dijon on the bottom half of the buns, top with burgers and bun tops.
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