For the flavor base: Add 2 tablespoons oil and the chopped mushrooms to a large pan with straight sides over medium heat. Cook, stirring, until the mushrooms deepen in color and reduce, releasing most of their moisture, 10 to 12 minutes. Remove from the pan with a slotted spoon to a paper towel-lined plate, then season with salt and pepper. Lower the heat to medium-low and add the remaining 2 tablespoons oil along with the garlic, ginger and red chile flakes. Cook, moving around the pan, until a waft of aroma from the pan smells like red chile flakes. Remove from the heat and add the fish sauce, lemon zest and juice, a pinch of salt and several grinds of black pepper. Let cool completely.
For the London broil: Sprinkle the meat with salt and pepper. Rub half of the flavor base all over the meat. Wrap tightly in plastic wrap and refrigerate 4 hours; refrigerate the remaining flavor base.
Bring the meat to room temperature 2 hours before grilling. Add the remaining flavor base and the mushroom caps to a small food processor. Pulse until pulverized, then add the butter and lemon zest. Pulse again to combine, then add the cilantro and pulse again. Transfer to a small saucepan.
Preheat a grill for direct and indirect cooking on medium-high heat. Remove the wrapping from the meat and place the saucepan on the direct-heat portion of the grill to heat. Drizzle the mushrooms with olive oil and sprinkle with salt and pepper. Place the mushrooms and the meat over the direct heat of the grill and sear both for about 6 minutes, then flip. Continue to cook the steak about another 6 minutes for medium-rare; meanwhile, move the mushrooms to the indirect side of the grill to finish cooking. Remove the mushrooms and sauce when the steak comes off the grill.
Rest the meat for about 5 minutes, then slice 1/4- to 1/2-inch thick. Cut the caps into quarters and toss in a bowl with a bit of the melted compound butter and another squeeze of lemon juice. Serve warm with compound butter alongside.