Preheat the oven to 475 degrees .
Halve the squash; scoop out the seeds. Drizzle each half with oil and season with salt and pepper. Arrange, cut side down, on a nonstick baking sheet. Roast until just tender, 18 to 22 minutes.
In a soup pot with a lid, heat 2 tbsp. oil, two turns of the pan, over medium-high. Add the onion, celery, garlic, ginger, coriander, cumin, garam masala and paprika. Cover and cook, stirring occasionally, until the vegetables are tender, 8 to 10 minutes. Add the stock and bring to a boil.
Heat a griddle pan or large skillet over medium. Lightly butter one side of each slice of bread; spread the unbuttered side of 4 bread slices with the mango chutney. Build the sandwiches with the bread, buttered sides out, and the cheese. Griddle until the bread is crispy and golden and the cheese melts, about 2 minutes per side. Cut each sandwich into 1-inch-wide sticks.
Using an oven mitt or kitchen towel to hold the squash, scoop the flesh into the soup pot with the onion mixture. Using an immersion blender or working in batches in a blender, puree the soup; season. Serve with the grilled cheese sticks for dunking.