Roast the spaghetti squash. Preheat the oven to 400°F (200°C).
Halve the squash lengthwise and scoop out the seeds. Drizzle the cut sides with olive oil, season with a pinch of salt and pepper, and place cut side down on a parchment-lined baking sheet. Roast for 35–45 minutes, until the strands pull apart easily with a fork and the edges are lightly caramelized.
Shred the squash. When cool enough to handle, use a fork to scrape the flesh into long “spaghetti” strands.
Toss with a little olive oil and a pinch of salt to keep it from drying out. Set aside.
Season and sear the chicken. Pat the chicken dry. Season with 1/2 teaspoon salt, pepper, and Italian seasoning.
Heat 1 tablespoon olive oil (or the reserved sun-dried tomato oil for extra flavor) in a large skillet over medium-high heat. Sear the chicken in a single layer until golden and cooked through, about 5–7 minutes. Transfer to a plate.
Sauté aromatics. In the same skillet, reduce heat to medium and add the remaining 1 tablespoon oil if needed.
Add garlic and sun-dried tomatoes. Cook for 30–60 seconds until fragrant, stirring so the garlic doesn’t brown.
Build the sauce. Pour in the chicken broth to deglaze, scraping up any browned bits. Simmer for 2 minutes to reduce slightly.
Stir in the cream, bring to a gentle simmer, and cook for 2–3 minutes until it thickens enough to coat a spoon.
Add cheese and greens. Lower the heat and stir in the Parmesan until smooth. Add the spinach or kale and let it wilt. Season with the remaining 1/2 teaspoon salt, a pinch of red pepper flakes (if using), and the lemon zest and juice.
Combine chicken and sauce. Return the cooked chicken and any juices to the skillet.
Toss to coat and simmer 1 minute to marry the flavors. Taste and adjust salt, pepper, and lemon.
Assemble. Add the spaghetti squash strands to the skillet, gently tossing to soak up the sauce. If your skillet isn’t large enough, combine in a big bowl.
Finish with chopped fresh basil and an extra sprinkle of Parmesan.
Serve. Spoon into bowls. For a restaurant touch, drizzle with a little olive oil and add a few extra slivers of sun-dried tomato on top.