Preheat the oven to 465 degrees F (425 degrees F convection).
Mix the squash with 2 tablespoons of the olive oil, 2 teaspoons of salt, and a good grind of black pepper. Spread out on a large parchment-lined baking sheet and roast for 35 to 40 minutes, until golden brown. Set aside to cool.
Reduce the oven temperature to 340 degrees F or 300 degrees F for a convection oven.
Place the tomato halves on a parchment-lined baking sheet, skin side down. Sprinkle with 1/4 teaspoon of salt, drizzle with the remaining tablespoon of oil, and cook for 80 minutes, until softened.
Place the ginger, chile, garlic, sugar and ΒΌ teaspoon of salt in a medium bowl. Mix to form a paste, then spoon this on top of the tomatoes. Cook for another 40 minutes, until the tomatoes are caramelized, and set aside to cool.
Place all the ingredients for the lime yogurt in a small bowl, with 1/2 teaspoon of salt and a good grind of black pepper. Mix well and keep in the fridge until ready to serve.
Spread the squash out on a large platter and layer the tomatoes in between. Drizzle over the lime yogurt, sprinkle with the cilantro, cashews, and shallots, and serve.