Preheat the oven to 400°F (200°C). Line a baking sheet with foil and lightly spray or oil it.
Slice each spaghetti squash in half lengthwise and scoop out the seeds. Brush the cut sides with about 1 tablespoon olive oil and season with salt and pepper.
Place the squash halves cut-side down on the prepared baking sheet and roast for 25 minutes, until the flesh is tender when pierced.
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 for about 2 minutes until fragrant.
Add the broccoli florets to the skillet and cook, stirring frequently, for 6–8 minutes until slightly softened.
Pour in the milk and heat for 1–2 minutes until warmed through, then stir in 1 cup of the grated cheddar until melted and combined; remove from heat.
Remove the roasted squash from the oven and flip each half cut-side up. Use a fork to loosen some of the squash strands and season with additional salt and pepper if desired.
Divide the broccoli-cheddar mixture evenly among the squash halves. Sprinkle the remaining 1 cup cheddar over the tops, then evenly top with the seasoned panko breadcrumbs.
Return the baking sheet to the oven and bake for 10–15 minutes more, until the cheese is bubbly and the breadcrumbs are golden brown.