Preheat the oven to 375°F

Set the squash halves cut-side up on a baking sheet.
Brush the interior of each half with olive oil and season
well with salt and pepper. Roast until the flesh is
tender when pierced with a fork, about 30 minutes.
Remove from the oven and cool.

Heat 1 tablespoon olive oil in a large skillet over medium heat. Add the sausage and cook, breaking up with a spoon into small crumbles, until browned
and cooked through, about 5 minutes. With a slotted spoon, transfer the
sausage to a paper towel-lined plate to drain. Pour all but 2 tablespoons
of the oil in the skillet into a small bowl and discard. Add the scallions and
garlic to the skillet and cook until softened, 1 to 2 minutes.

Add the mustrooms, season with salt and pepper, and cook, stirring, until the mushrooms
no longer release any liquid and are brown, 6 to 8 minutes. Transfer the vegetables to a medium bowl and cool. Add the cooked rice to the mushrooms along with the sausage, scallion
greens, half the parsley, thyme, and shredded Parmesan and stir well to
combine. Taste and season with salt and pepper.

Whisk the eggs into the
chicken stock, pour over the rice mixture, and stir until well combined. Fill
each squash half with one-fourth of the stuffing and transfer them to the
oven. Bake until hot and the cheese is melted, 12 to 14 minutes.Let stand 10 minutes before garnishing with the remaining parsley and serving.