MAKE THE ONIONS:
Heat butter in a large skillet over MED heat. Add sliced onions and stir to coat in melted butter. Sprinkle in salt, pepper, and thyme and stir to combine. Saute, stirring occasionally, for about 7-10 minutes, until onions are soft and golden brown at the edges.
Pour in half the Guinness, cover, and cook until liquid is about 3/4 of the way evaporated. Repeat with remaining Guinness. This time, cook until liquid is nearly all evaporated. Remove onions from pan and set aside.
MAKE THE GLAZE:
To a small saucepan, add whiskey and bring to a low boil. Continue to boil for about 2 minutes, then add remaining glaze ingredients and whisk to combine.
Reduce heat to LOW and simmer for 5-7 minutes, until glaze is slightly thickened.
Remove from heat and set aside.
MAKE THE BURGERS:
To a large mixing bowl, add ground beef, salt, pepper, and dried minced onion. Use your hands to mix it all together. Separate the meat mixture into 4 equal portions.
Shape meat into flattened balls, place about 1 Tbsp blue cheese crumbles in the center of the ball and push into the ball. Mold the meat around the cheese and form into a patty.
Heat a large cast iron skillet or grill pan over MED heat. Drizzle a bit of olive oil in the pan. Cook burgers about 6 minutes per side, or until desired doneness. The last minute of cooking, brush burgers with prepared whiskey glaze and place a slice of cheese on the burgers, then cover pan loosely with foil.
Remove burgers to a plate and cover with foil for 3 minutes.
Line bottom bun with arugula lettuce, then top with burger. Place a generous helping of caramelized onions on top of the burger, then drizzle with additional glaze. Top with top bun and enjoy!