Step 1

Process mushrooms in food processor until smooth paste forms (paste will resemble thick oatmeal), scraping down sides of bowl as needed, about 1 minute. Transfer to large bowl and cover. (Do not wash out processor bowl.)

Step 2

Microwave mushrooms until liquid released begins to boil, about 3 minutes, stirring halfway through. (Do not walk away during final minute; mushrooms could boil over.) Transfer mushrooms to large fine-mesh strainer set over bowl. Using spatula, press on mushrooms to extract ½ cup liquid (if more than ½ cup is removed, stir extra liquid back into mushrooms). Discard liquid and return mushrooms to bowl. Refrigerate mushrooms until room temperature, about 20 minutes.

Step 3

Return mushrooms to processor bowl. Add beef and salt and process until mixture is uniform and begins to pull away from sides of bowl, about 20 seconds. Divide mixture into 4 equal portions and shape into patties that are 4½ inches in diameter. Sprinkle both sides of each patty with pepper. (Patties can be refrigerated overnight or tightly wrapped and frozen for up to 1 month; if frozen, thaw before cooking.)

Step 4

Heat oil in 12-inch skillet over medium-high heat until shimmering. Transfer patties to skillet and cook until well browned on both sides and burgers register 135 degrees (for medium-rare) or 155 degrees (for medium-well), 6 to 10 minutes. If using cheese, place 1 slice on each burger 1 minute before burgers finish cooking. Transfer burgers to plate and let rest for 5 minutes, then transfer to buns and serve.