In a large, heavy pot, melt the butter over medium-high heat. Add the onions and cook, stirring, until soft, about 4 minutes. Add the garlic, curry powder, salt, turmeric, cumin, and cayenne, and cook, stirring, until fragrant, 45 seconds. Add the squash and cook, stirring, for 3 minutes. Add the stock and bring to a boil. Reduce the heat, cover, and simmer, stirring occasionally, until the squash is soft, 25 to 30 minutes. Remove from the heat.
With a hand-held immersion blender, puree on high speed. Add the cream, stir, and adjust the seasoning, to taste. Cool to room temperature, then refrigerate until well chilled.
Preheat the fryer to 350 degrees F.
Fry the pappadams in batches until golden and crisp, about 1 1/2 minutes each. Drain on paper towels. To serve, ladle the soup into shallow bowls. Sprinkle the cucumbers and cilantro over the top and garnish with pappadams.