Steam/cook broccoli (fresh or frozen) in a saucepan on medium heat until broccoli is fork-tender.
In a blender of food processor, pulse walnut pieces until they resemble a course meal, but have not yet become walnut butter. Place walnut pieces in a large mixing bowl.
In a blender or food processor, pulse cooked/drained broccoli until it is crumbled into very tiny pieces, but not yet a puree (eeeeeewwww).
Please broccoli and remaining ingredients in mixing bowl with walnuts and rice. If your mixture seems dry and crumbly, go ahead and add a tablespoon of water. This stuff can turn into soggy goop pretty quickly, so you'll want to be very careful to add only a scant tablespoon og water at a time to your burger mixture.
Stir burger mix well, then shape into 4 equal burgers.
Pour a tablespoon or 2 of oil (just enough to coat the pan) in a skillet and place on medium to medium-high heat.
Gently pan-fry your burgers, allowing a few minutes for cooking on each side.
You'll know they are done when the burger surface has a nice dark brown crispiness.
Serve burgers warm on vegan buns with veggies and condiments of your choice (I chose a mix of Just Mayo, pickle relish, and ketchup.)