Preheat the oven to 400°F and line a baking sheet with foil; spray or lightly oil the foil.
Slice each spaghetti squash in half lengthwise and scrape out the seeds. Brush the cut sides with half of the olive oil and place cut-side down on the prepared baking sheet.
Roast the squash in the preheated oven for 25 minutes, or until the flesh is tender when pierced with a fork.
While the squash roasts, heat the remaining olive oil in a skillet over medium-low heat. Add the diced shallot and minced garlic and cook about 2 minutes until softened and fragrant.
Add the broccoli florets to the skillet and sauté, stirring often, for 6–8 minutes until slightly softened.
Pour in the milk and warm for 1–2 minutes, then stir in 1 cup of the grated white cheddar; remove from heat and stir until the cheese melts and the mixture is combined.
Remove the roasted squash from the oven and flip the halves cut-side up. Use a fork to loosen some of the squash strands, season with salt and pepper, then evenly divide the broccoli-cheddar mixture among the squash halves.
Sprinkle the remaining 1 cup of cheddar over each filled squash, then top with the seasoned panko breadcrumbs.
Return the stuffed squash to the oven and bake 10–15 minutes until the cheese is melted and golden and the breadcrumbs are lightly browned.
