Using a damp paper towel, gently wipe the mushrooms clean. Remove the stem, and use a spoon to gently scrape out and discard the black gills.
Whisk the olive oil and vinegar to combine. Pour into a large resealable bag, then add the garlic and mushroom caps. Seal the bag and toss to cover the mushrooms with the liquid. Refrigerate for 30 minutes or up to overnight, turning the bag a few times to redistribute the marinade.
Heat the oven to 350 degrees F and line a baking sheet with aluminum foil. Toss the butternut squash with a light coating of olive oil, then sprinkle with salt. Spread in a single layer on a baking sheet and cook until fork tender, around 20 minutes.
Remove mushrooms from the marinade, saving the liquid, and place cap-side down on a foil-lined baking sheet. Roast for 15 minutes.
While the mushrooms are roasting, combine the farro, squash, shallot, cranberries, pine nuts, arugula, salt and pepper in a large bowl with two tablespoons of marinade.
Remove the mushrooms from the oven, carefully pouring off any liquid. Distribute the filling evenly between the mushroom caps (there may be leftover filling if the mushrooms are very large). Cook for an additional 5 minutes, then serve.