Chop mushrooms to ¼-inch dice (0.6 cm) and put them in your cast iron skillet. Dry saute them for approximately 10 minutes. You want the liquid from the mushrooms to be released, and the mushrooms to be reasonably dry.
Put the raw ground beef, steak seasoning, and sauteed mushrooms in a large bowl. Mix these ingredients together thoroughly with clean hands.
Shape the beef mixture into three thin and flat burger patties that are about 4 inches (10 cm) in diameter and ½-inch thick (1.27 cm). Put the patties in a single layer on a plate. Wash your hands both before and after touching the raw burger.
Let the plate of raw burgers sit in the fridge for a minimum of four hours or a maximum of overnight. The drying will help the smoke adhere to the burgers.
Set up your stovetop smoker. For my Nordic Ware kettle smoker, I added 2 tablespoons of mesquite shredded wood chips to the bottom of the smoker. The drip tray goes in the smoker next, and then the smoker tray.
About one hour before you are ready to eat, gently place your burgers in a single layer on the smoker tray. Then put the lid on the smoker. Make sure that the thermometer is attached to the smoker correctly and that your stove vent is on. Turn the heat on your stovetop to medium-low.
Allow the smoker to come to 190°F.
While the temperature on the smoker rises, make your caramelized onions. Put the olive oil and thinly sliced sweet onions in a cast-iron skillet. Allow the onions to slowly turn golden brown on the stovetop over medium-low heat. Stir the onions often. Getting your onions caramelized will take around 40 minutes.
Once the smoker temperature reaches 190°F (88°C), turn the stovetop to low. Set a timer to smoke the burgers for 15-20 minutes. The smoker needs to stay in the 190°F-210°F (88°C-99°C) range for the entire smoking period. (I usually smoke for 15 minutes.)
If the heat on your smoker is rising too rapidly, quickly lift the lid and peek at your burgers. This is an easy way to adjust the temperature quickly.