Heat a grill to medium high. The grill is ready when you can hold your hand over the grates (about 1 foot above) for no more than a 5-second count.
Add the mushrooms to the grill, gill-side up. Sprinkle each cap with salt. Close the hood and grill for 8 to 10 minutes.
Meanwhile in a large bowl, combine the ketchup, mayo, mustard, and pickle juice. Mix well and set aside close to the grill.
After about 10 minutes, open the grill. Liquid will have accumulated in the cap. Using grill tongs, gently pour the liquid off each mushroom into the ketchup mixture. Return each mushroom to the grill, gill side down.
Mix the ketchup mixture to incorporate the mushroom liquid. Using the tongs, dunk each mushroom in the ketchup mixture until generously coated and return to the grill, gil-side up. Grill uncovered for 2 minutes. Flip the mushrooms and cook 2 minutes more.
Add slices of cheese to the mushrooms. Place buns on the top rack of the grill to toast. Close the hood and grill for 1 to 2 minutes or until the cheese is bubbly and melted and buns are warm.