Preheat the oven to 400°F (200°C).
Slice the spaghetti squash in half lengthwise and remove the seeds using a spoon.
Brush the flesh of the squash with olive oil and season with a pinch of salt and pepper. Place cut-side down on a parchment-lined baking sheet and roast for 40 to 45 minutes until tender.
Remove squash from oven and allow to cool slightly. Use a fork to scrape the flesh into long, spaghetti-like strands, then transfer the strands to a large mixing bowl.
In a small saucepan over medium heat, melt the butter. Add the minced garlic and sauté for 1 minute until fragrant.
Whisk in the heavy cream and thyme. Simmer for 3 to 5 minutes until the liquid reduces slightly and coats the back of a spoon.
Pour the cream mixture over the squash strands. Fold in 1/2 cup of the Gruyère and 1/4 cup of the Parmesan.
Transfer the mixture to a buttered 2-quart baking dish. Spread evenly.
In a small bowl, combine the remaining cheeses with the Panko breadcrumbs. Sprinkle this mixture over the top of the squash.
Bake for 15 minutes at 400°F (200°C), or until the top is golden brown and the cream sauce is bubbling.