Brush 1 tablespoon oil on cut side of squash halves; season with ½ teaspoon salt and ¼ teaspoon pepper.
Place squash, cut side down, on a parchment-lined baking sheet; bake until slightly tender, 20 to 25 minutes. Remove from oven; turn cut side up.
Meanwhile, heat 1 tablespoon oil in a large skillet over medium-high. Add mushrooms and shallot; cook, add 2 oz. white wine stirring occasionally, until tender, about 5 minutes. Add spinach; cook until wilted, about 1 minute. Season with ¼ teaspoon salt and ⅛ teaspoon pepper. Remove from heat and fold in bread and goat cheese.
Divide mushroom mixture among squash halves, return to oven, and bake until tender, 8 to 10 minutes. Increase temperature to high broil. Broil until lightly browned, 3 to 5 minutes.