How to Make It
In a saucepan, bring the vinegar, sugar, salt and chile to a boil. Remove from the heat and let cool. Add the lime juice and pour the brine over the cucumber slices. Press the cucumber slices down into the liquid with a small plate and place a can on the plate; let stand at room temperature for 4 hours.
In a bowl, combine the ketchup with the mayonnaise, cornichons, shallots, vinegar, miso and Sriracha and season with salt.
Light a grill. Season the patties with salt and black pepper and grill over high heat, turning once, until the burgers are charred outside and medium-rare within, about 6 minutes.
Spread the Russian dressing on the rolls and cover with the lettuce and tomatoes. Top with the burgers and pickles; serve.