Make sure to process the mushrooms just until coarsely chopped (about 4 pulses) so they match the texture of the ground beef. The mushroom mixture will look a little wet in the pan at first; take the time to let as much liquid as possible cook out for a richer patty that will hold its shape when seared in the pan.
1/2 cup white vinegar
1 tablespoon chopped fresh dill
1 teaspoon kosher salt
1 medium English cucumber, cut into 1/8-inch-thick slices (about 2 1/2 cups)
To prepare pickles, combine vinegar, dill, salt, and cucumber in a large zip-top plastic bag or jar; seal. Turn bag until ingredients are well combined. Refrigerate at least 1 hour, turning bag or shaking jar occasionally.
To prepare burgers, place cremini and button mushrooms in the bowl of a food processor; pulse 4 to 5 times or until coarsely chopped. Heat a large skillet over medium-high heat. Add oil to pan; swirl to coat. Add onion and garlic; sauté 3 minutes. Add mushrooms, 1/4 teaspoon salt, and 1/8 teaspoon pepper to pan; cook 7 to 10 minutes or until liquid evaporates, stirring occasionally. Remove pan from heat; cool mushroom mixture completely.
Place beef in a bowl. Add mushroom mixture to beef; stir just until combined. Refrigerate 10 minutes. Shape beef mixture into 4 (1/2-inch-thick) patties. Sprinkle evenly with remaining 1/4 teaspoon salt and remaining 1/8 teaspoon pepper.
Heat a large cast-iron skillet over medium-high heat. Coat pan with cooking spray. Add patties to pan; cook 2 minutes on each side or until desired degree of doneness.
Combine mayonnaise, ketchup, and mustard in a small bowl, stirring with a whisk. Spread mayonnaise mixture evenly over bottom and top halves of buns. Place 4 pickle slices on bottom half of each bun (reserve remaining pickles for another use). Top evenly with patties, lettuce, tomato, onion, and top halves of buns.