Consider making the sauce ahead of time, even days ahead. (See the sauce directions below.) Clean the portobello mushrooms. Slice off the stems so that the mushrooms are flat. (You can food process the stems, or chop them up and put a little on top of each mushroom so as not to waste them.) Put them on a plate. Season the mushrooms with salt and pepper. Put the garlic, ginger, sesame oil, and soy sauce over each portobello. It's not an exact science; just make sure each mushroom gets about the same amount of each ingredient. Heat the grill to medium high, between 375-400 degrees.
Brush the pineapple slices with olive oil on both sides.
Spray your grill with olive oil if possible. When it reaches at least 375 degrees, grill the mushrooms and the pineapple. Leave the mushrooms cap side up, and do not flip them so that you can retain all of the juices. The mushrooms will take app 7-9 minutes. (They should look juicy, not shriveled, although they will shrink in size.) Grill the pineapples for about 3 minutes on each side, until you can see grill marks. When you can tell that the mushrooms are 1-2 min. away from being done, add the Havarti on top of the mushrooms, and grill until melted. At the same time, add your brioche buns and toast for app 2 minutes. Brioche burns fast, so keep a close eye! You might want to keep it on the top rack away from direct heat.
Serve with thinly sliced red onion, arugula, pickled jalapeños, and jalapeño sauce (instructions below).