Roast the squash: Preheat your oven to 400°F (200°C). Halve each spaghetti squash lengthwise and scoop out the seeds. Rub the cut sides with 1 tablespoon olive oil and season with salt and pepper.
Place cut-side down on a lined baking sheet and roast 35–45 minutes, until the strands pull away easily with a fork and the edges are lightly caramelized.
Cook the bacon: While the squash roasts, add chopped bacon to a large skillet over medium heat. Cook until crisp, 7–10 minutes. Transfer bacon to a paper towel–lined plate.
Reserve 1–2 tablespoons bacon fat in the pan, discarding the rest.
Sauté aromatics: In the same skillet, add the remaining 1 tablespoon olive oil if needed. Sauté the onion with a pinch of salt for 3–4 minutes until soft. Stir in garlic, smoked paprika, and Italian seasoning; cook 30–60 seconds until fragrant.
Make it creamy: Reduce heat to low.
Stir in the chicken broth and cream cheese, breaking the cream cheese up with a spatula until smooth. Add the Greek yogurt (or sour cream) and 3/4 cup shredded cheese. Cook gently, stirring, until the mixture is creamy and combined.
Add chicken and bacon: Stir in the cooked chicken and half of the bacon.
Taste and season with salt, pepper, and red pepper flakes if you like a little kick.
Shred the squash: When the squash is cool enough to handle, flip the halves cut-side up and use a fork to pull the strands into “noodles,” leaving a 1/4-inch border so the shells stay sturdy.
Combine: Add the squash strands to the skillet and toss until everything is evenly coated. If the mixture seems too thick, splash in a little more broth; if too thin, let it simmer for a minute to tighten up.
Load the boats: Return the mixture to the squash shells. Top with the remaining shredded cheese and Parmesan.
Sprinkle over the remaining bacon.
Bake: Reduce oven to 375°F (190°C). Place the stuffed squash on a baking sheet and bake 10–15 minutes, until the cheese is melted and bubbly with golden spots.
Finish and serve: Let rest 5 minutes. Top with sliced green onions and serve hot right from the shells.