Preheat oven to 400°. On a rimmed baking sheet, toss squash, 1 tbsp. oil, 1/2 tsp. salt, and the pepper. Roast until squash is tender when pierced, about 30 minutes; let cool.
Meanwhile, arrange bread slices on a second rimmed baking sheet and toast in oven until golden brown, 8 to 10 minutes; let cool.
In a food processor, pulse arugula, almonds, remaining 1/2 cup oil, remaining 1/2 tsp. salt, and the shredded parmesan, scraping down bowl as needed, until a rough paste forms.
Cut squash into 1/2-in. pieces; you should have 1 1/2 cups (save any extra for other uses).
Spread toasts with arugula pesto, then top each with a few cubes of roasted squash and a sprinkle of shaved parmesan.