Remove brown gills from the undersides of mushrooms using a spoon; discard gills.
Melt butter in a large skillet over medium heat; add leeks and garlic, and sauté until tender.
Remove from heat, and let cool.
Stir together avocados and next 3 ingredients in a medium bowl; stir in leek mixture.
Divide and press goat cheese evenly into mushroom caps; top evenly with avocado mixture.
Sprinkle evenly with chopped walnuts, and drizzle with olive oil.
Place on a rack in a broiler pan.
Bake mushrooms at 400° for 5 minutes; cover loosely with aluminum foil, and bake 5 more minutes or until walnuts are golden brown and mushrooms are tender.