Preheat the oven to 375 degrees F (190 degrees C).
Brush cut sides of squash with 1 tablespoon olive oil and place, skin-side up, on a baking sheet.
Bake in the preheated oven until tender and lightly browned, about 40 minutes.
While squash bakes, heat 1 tablespoon olive oil in a skillet over medium heat; stir in mushrooms, onion, garlic, and thyme. Cook and stir until soft, about 5 minutes. Add rice; cook and stir until rice is coated, about 1 minute. Pour in chicken broth and bring to a boil. Reduce heat to low, cover, and simmer, stirring occasionally, until rice is tender and liquid has been absorbed, about 15 minutes. Remove from the heat and stir in Cheddar cheese.
Remove squash from the oven, flip over, and season with salt and pepper. Fill squash with rice mixture, sprinkle with panko, and drizzle with remaining olive oil.
Return to the oven and bake until heated through, about 10 minutes. Turn on the broiler and broil until panko is brown and crispy, 2 to 3 minutes.