In a large pot over medium heat, add the oil and coriander and/or cumin and cook, stirring often, until fragrant, about 1 minute.
Stir in the squash, mushrooms, farro, water, and salt and stir to combine. Cover and bring to a boil over high heat, then reduce the heat to medium-low and simmer, covered, rapidly for about 30 minutes, or until the farro is tender, stirring halfway through. If your farro takes longer to cook and the liquid is reducing too much, just stir in some more water.
Stir in the greens, submerging them into the liquid, and cook until wilted, about 1 minute. Remove from heat.
Add your miso paste to a heat-proof bowl and add at least ½ cup of hot liquid from the pot to that bowl. Transfer a few pieces of squash to the same bowl (helps to thicken the soup). Use a fork to mash the miso/squash until it dissolves. Stir back into the pot along with the ginger and lemon. You can start with 1 tablespoon of each ginger and lemon and add more, to taste.
Serve topped with a fried egg (if using) and fresh black pepper.