Preheat oven to 400°F (200°C). Line a baking sheet with foil and lightly spray with nonstick spray.
Slice each spaghetti squash lengthwise and scoop out the seeds. Brush the cut sides with about half of the olive oil and place cut-side down on the prepared baking sheet.
Roast the squash for 25 minutes, 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 fragrant and softened.
Add the broccoli florets to the skillet and cook, 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 cheddar until melted and combined; remove from heat.
Remove the squash from the oven and flip each half cut-side up. Use a fork to scrape some of the squash flesh to create a bowl-like texture. Season the squash with salt and pepper.
Divide the broccoli and cheddar mixture evenly among the four squash halves. Sprinkle the remaining 1 cup of cheddar over each, then top with the seasoned panko breadcrumbs.
Return the baking sheet to the oven and bake 10–15 minutes more, until the cheese is melted and golden and the breadcrumbs are toasted.
Remove from oven and let cool a few minutes before serving.