In a skillet, sweat the shallots without browning. Add the apricots, honey and apricot nectar. Simmer until the liquid has evaporated. Set aside to cool at room temperature, then stir in the pistachios and a dash of vinegar. Refrigerate.
In a large bowl, combine all the ingredients for the burgers. Shape into 4 patties and refrigerate.
Preheat the barbecue to high and when the grill is very hot, place the patties on the grill and sear them on both sides, making sure not to turn them too soon. When the burgers are well marked and can be lifted off the grill easily, remove them to another part of the barbecue where the heat has been reduced and let them finish cooking undisturbed.
While the burgers are cooking, brush the mushroom caps with oil and grill on both sides until done. Set them aside.
Turn off one of the burners, place the burgers on that part of the grill and top each with a slice of cheese. Close the lid for a few minutes and cook over indirect heat until the cheese is melted.
Serve in warmed or toasted buns. Top each burger with a mushroom cap, a generous spoonful of chutney and some cucumber slices.