Preheat the oven to 350 degrees F (175 degrees C). Line an extra-large baking sheet with foil.
Cut each squash in half, slicing through stem and blossom ends. Remove seeds. Prick skin with a paring knife. Brush 1 tablespoon oil on cut sides and sprinkle with 1/4 teaspoon each salt and pepper. Arrange, cut-sides down, on the prepared baking sheet.
Bake in the preheated oven until tender, 45 to 50 minutes. Reduce oven temperature to 300 degrees F (150 degrees C).
Meanwhile, rinse and drain farro, then transfer to a saucepan. Stir in broth and 1/4 teaspoon salt. Bring to a boil over high heat. Reduce heat to medium-low and simmer, covered, until tender, about 20 minutes. Drain any excess broth.
Heat remaining 2 tablespoons oil in a large skillet over medium heat. Add mushrooms and onion; cook until mushrooms have softened and onion is translucent, 8 to 10 minutes. Add wine; cook, stirring, until reduced by half, about 30 seconds.
Add cranberries and thyme; cook, stirring, until cranberries have plumped, about 4 minutes more. Remove thyme. Stir in cooked farro, pecans, and parsley. Heat through. Season with remaining salt and pepper and balsamic vinegar. Spoon farro mixture into squash halves. Return to oven to keep warm until serving.