Heat ½ tablespoon of the olive oil in a medium saucepan. Add the farro and cook over medium heat, stirring, until fragrant, 2 minutes. Add 1½ cups of water and a pinch of salt, and bring to a boil. Cover and simmer for 20 minutes. Check the farro and continue to simmer for another few minutes until the farro is chewy but not mushy. Remove the pan from the heat and drain off any remaining water.
In a large, deep skillet, heat the remaining 2½ tablespoons of olive oil over medium heat. Add the onion and cook until translucent, 5 to 8 minutes. Add the carrots and garlic and cook until fragrant, 2 minutes. Add the mushrooms and cook until softened and cooked through, another 3 to 4 minutes. Set the mushroom and vegetable mixture aside to cool.
In a food processor or blender, pulse 1 cup of the black beans to a chunky puree. Transfer to a large mixing bowl. Mix in the remaining black beans, cooked farro, cooled mushroom and vegetable mixture, panko bread crumbs, edamame, cilantro, parsley, mustard, soy sauce, chili powder, cumin, salt, and pepper. Allow to cool until the mixture is no longer steaming. Taste and adjust the seasonings to your liking. Whisk the eggs in a small bowl and stir them into the vegetable mixture. Cover and refrigerate the burger mixture for at least 30 minutes or for up to a day.
When ready to cook the veggie burgers, heat the oven to 400°F. Line a baking sheet with parchment or some other nonstick liner.
Form handfuls of the mixture into patties roughly 4 inches in diameter and ¾ inch thick. Lay them on the baking sheet spaced an inch or two apart. Bake the burgers in batches if all the patties don’t fit; alternately, form some patties and freeze them for later (see below).
Bake the burgers for 20 to 25 minutes, until the tops are dry and the patties feel firm around the edges. If desired, top the burgers with cheese and run them under the broiler for 30 to 60 seconds until the cheese has melted. Serve the patties warm on rolls with ketchup, mustard, mayo, and any other favorite toppings.
Baked patties will keep in the refrigerator for up to 1 week in an airtight container. Unbaked patties can be frozen for up to 3 months: freeze the shaped patties on a parchment-lined baking sheet until solid, then stack the burgers between layers of wax paper and seal inside a freezer proof container. Increase the baking time by 5 to 10 minutes for frozen patties.