Triple Decker Burgers with Roasted Vegetables and Cheese Sauce

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Vegetables:

2 medium carrots, roughly chopped

2 large stalks celery, roughly chopped 

2 medium red onions, roughly chopped 

2 tablespoons canola oil 

Kosher salt 

2 large dill pickles, chopped, plus 2 tablespoons pickle juice 

3 tablespoons ketchup 

Cheese Sauce:

1 cup heavy cream

1 tablespoon Dijon mustard 

1 tablespoon low-sodium soy sauce 

1 teaspoon hot sauce, preferably Tabasco 

1 large clove garlic, grated 

1 cup shredded sharp Cheddar

6 slices American cheese, torn in pieces 

Burgers:

1 1/2 pounds ground beef (preferably chuck or sirloin)

Kosher salt and freshly ground black pepper

1/4 cup canola oil

4 seeded hamburger rolls, toasted 

1 small red onion, sliced thinly and soaked in ice water