Heat 2 Tbsp. oil in a medium Dutch oven over medium-high.
Cook mushrooms, tossing occasionally, until browned and crisp, 6–8 minutes.
Season with salt; transfer to a plate.
Let pot cool 2 minutes.
Set pot over medium heat and pour in remaining 3 Tbsp. oil.
Cook onion and shallots, stirring occasionally, until softened and starting to brown, 6–8 minutes.
Add sliced garlic and butter; cook, stirring often, until garlic is softened and starting to brown, about 2 minutes.
Add wine; simmer, stirring once, until reduced by half, about 5 minutes.
Add farro, miso, and 6 cups water. Bring to a boil, stirring to dissolve miso.
Reduce heat and simmer, uncovered, stirring once or twice, until farro is halfway cooked, about 20 minutes.
Taste broth and season with salt.
Add squash (submerge the best you can), cover, and cook until tender, about 30 minutes.
Add mushrooms; cook just until heated through, about 2 minutes.
Ladle stew into bowls; top with parsley, lemon zest, and chopped garlic.