Preheat oven to 400°F. Place squash on a rimmed baking sheet; toss with 1 tablespoon oil. Bake until tender, 20 to 25 minutes. Remove squash. Reduce oven temperature to 375°F. Place a wire rack in a large baking sheet.
Combine squash, goat cheese, butter, pepper, and 1/4 teaspoon salt in a food processor; process until smooth. Combine panko, thyme, and remaining 1 1/2 tablespoons oil and 1/4 teaspoon salt in a small bowl.
Spoon about 2 teaspoons squash mixture into each mushroom cap. Press about 1 teaspoon panko mixture into squash mixture. Place mushrooms in a single layer on rack on baking sheet. Bake until panko is toasted, about 15 minutes.