Wipe the mushrooms clean with a damp paper towel. Twist off the stem and use a spoon to gently remove the black gills without breaking the mushroom.
Whisk together extra virgin olive oil and balsamic vinegar. Pour into a large sealable plastic bag and add the smashed garlic cloves and mushroom caps. Seal the bag and allow the marinade to cover the mushrooms. Refrigerate for at least 30 minutes or overnight, turning the bag periodically to redistribute the marinade.
Preheat the oven to 350 degrees. Toss the butternut squash in a small amount of olive oil and sprinkle with salt. Spread in a single layer on a baking sheet and cook until soft but not mushy, 15–20 minutes.
Remove mushrooms from the marinade (do not discard the liquid) and place them cap side down on a baking sheet. Roast for 15 minutes.
While the mushrooms are roasting, combine the butternut squash, farro, shallot, dried cranberries, pine nuts, arugula, salt, and pepper in a bowl with two tablespoons of the marinade.
After the mushrooms have cooked for 15 minutes, remove them from the oven. Carefully pour off any liquid that has pooled in the mushrooms.
Distribute the filling evenly between the mushroom caps (there might be leftover filling depending on the size of the mushrooms). Cook for another 5 minutes and then serve.