Rinse 1 cup quinoa in a fine-mesh sieve under cold running water until the water runs clear. In a saucepan bring 2 cups water to a boil, add the rinsed quinoa, reduce heat to low, cover, and simmer 12 to 15 minutes until the quinoa is tender and the water is absorbed. Drain any excess water, fluff the quinoa with a fork, and set aside to cool.
While the quinoa cooks, heat some of the 1/4 cup olive oil (from the 1/4 cup olive oil divided) in a large sauté pan over low heat. Add 1 large shallot, minced, and 1/2 teaspoon crushed red pepper flakes. Cook, stirring, about 3 minutes until the shallot is soft.
Add the 2 large portobello mushroom caps (stems removed and caps finely chopped) to the pan and cook, stirring often, about 3 minutes more until the mushrooms are softened. Remove the pan from the heat and transfer the mushroom mixture to a large bowl.
Add the cooled quinoa to the bowl with the mushroom mixture. Season with 1/8 teaspoon salt and freshly ground pepper to taste. Stir to combine and allow the mixture to cool until it is comfortable to handle.
Once cooled, add 1 1/2 cups bread crumbs and 1/2 cup cooked, mashed yam to the bowl. Knead or mix with your hands until fully incorporated. If the mixture is too loose to hold together, add more mashed yam a little at a time until it binds.
To check binding, form a very small test patty and fry it in a small amount of the reserved olive oil in a sauté pan until browned. If the test patty falls apart, return to step 5 and add more mashed yam; if it holds together, proceed.
Divide the mixture and press firmly with your hands to form 8 evenly sized patties.
Heat a thin layer of the remaining 1/4 cup olive oil in a large skillet or grill pan over medium heat. Cook the patties in batches as needed, 3 to 4 minutes per side, until golden-brown on both sides and heated through. Transfer cooked patties to a plate and keep warm while you finish the rest.
Serve the patties on lettuce leaves if substituting for buns (use 6 to 12 lettuce leaves as available). Divide the 3 tomato slices and 3/4 cup guacamole (or sliced avocado) among the patties as desired.