Caramelized Onions and Blue Cheese Burgers

Red Robin's Bleu Ribbon Burger was inspiration to find a similar recipe.  This one with caramelized onions is fantastic!

Ingredients for caramelized onions:

2 tablespoons olive oil
2 large red, Spanish or sweet onions, thinly sliced
2 garlic cloves, minced or thinly sliced (optional)
1 tablespoon fresh rosemary leaves
½ teaspoon kosher salt, or more to taste

Ingredients for burgers:

1½ to 1¾ pounds ground chuck
3 ounces blue cheese, crumbled, plus extra for topping
1 teaspoon kosher salt
½ to 1 teaspoon freshly ground black pepper

Preparation:

To make the caramelized onions:
Place a large skillet over low heat and when it is hot, add the oil. Add the remaining ingredients and cook, stirring only occasionally as necessary, until the onions are deeply browned and slightly gooey, about 40 minutes. If the onions dry out, add water, 1 tablespoon at a time. Use immediately or cover and refrigerate for up to 2 days. (Yield: ¾ to 1 cup)

To make the burgers:
Place the beef on a work surface and divide into 4 balls of equal size. Divide the blue cheese into 4 portions of equal size. Using your thumb or the back of a tablespoon, make a shallow hole in each portion of beef and fill it with the blue cheese. Shape the burger and carefully seal the opening.
Form each into a patty about ¾ to 1 inch thick. To ensure more even cooking, make a ½-inch indentation with your thumb in the center of the burger. Sprinkle with salt and pepper.
Prepare a grill to medium-high.  Grill burgers. Serve on bun with caramelized onions.