Caramelize the onions: 

Heat a pan over medium heat and add the butter and olive oil. Add the sliced onions and cook for 5 minutes until they soften. Add the brown sugar, then pour in the beer. Continue cooking over low heat for 15 minutes, stirring occasionally, until the onions turn golden and the beer evaporates. 

Cook the hot dogs: 

Cut each hot dog in half and grill them in a pan or on a grill until they get a light brown color on both sides. 

Warm the buns:

Spread a little butter on the buns and toast them in a pan or on a grill until they turn golden. 

Build the Hot Dog Burgers:

Spread mustard and ketchup on the top half of the bun. On the bottom half, place a slice of cheese, two hot dog halves per burger, add the caramelized onions, and a few arugula or lettuce leaves if desired. Place the top half of the bun on top. 

Serve and enjoy:

Serve warm with fries, chips, or a light salad. A delicious and easy-to-make combination!