Preheat the oven to 425°F (220°C).
Prepare the garlic: trim the top of the garlic head to expose the cloves and remove any loose papery skin; place on a piece of foil, drizzle about 1 teaspoon olive oil over the cut cloves, wrap in foil and set aside on a baking sheet.
Prepare the eggplants: halve each eggplant lengthwise and rub the cut sides with olive oil, coarse salt, and pepper; place cut-side down on the baking sheet with the wrapped garlic.
Roast garlic and eggplants for 30–45 minutes, until the garlic is golden and soft and the eggplants are tender.
While they roast, heat a large skillet over medium heat with a drizzle of olive oil. Add the sausage and break it into small pieces, cooking until browned, about 5 minutes. Transfer the browned sausage to a plate.
In the same skillet, add a little more olive oil and the chopped onion; cook, stirring occasionally, until softened and lightly caramelized, about 5–7 minutes.
Add the chopped red peppers, chopped zucchini/summer squash, and chopped thyme; cook, stirring occasionally, until the vegetables soften, about 5 minutes.
Return the cooked sausage to the skillet and add the diced tomatoes, balsamic vinegar, chopped basil, chopped oregano, and salt and pepper to taste; simmer the sauce for about 5 minutes to combine flavors.
When the garlic and eggplant are done, squeeze the roasted garlic cloves from their skins into a small bowl and mash with a fork. Stir in 1/4 cup chopped basil, the pinch of crushed red pepper flakes, and the 1/4 cup Parmesan to make a garlic mixture.
Flip the eggplant halves cut-side up and spread the garlic–Parmesan mixture evenly over each half; drizzle with a little more olive oil and return to the oven for about 5 minutes.
Spoon the sausage and vegetable tomato sauce into the center of each eggplant half, pressing the filling into the cavity, then top each with sliced mozzarella.
Bake the stuffed eggplants for 10–15 minutes, until the cheese is melted and the sauce is bubbly. Serve warm with extra fresh basil if desired.