Preheat the oven to 350°. In a medium bowl, toss the squash with 2 tablespoons of the olive oil and season with salt and pepper. Spread the squash on a baking sheet and roast for about 30 minutes, turning once, until tender and lightly browned.
In a large skillet, heat the remaining 1/4 cup plus 2 tablespoons of olive oil. Add the kale and cook until it is wilted, about 8 minutes. Add the garlic slices and cook until the kale is tender, about 3 minutes longer. Season the kale with salt and pepper. Add the squash and toss gently to combine.
Heat a cast-iron grill pan. Brush the bread with olive oil and grill over high heat, turning once, until toasted. Mound the squash and kale on the toasts, top with the shaved cheese and serve.
The recipe can be prepared through Step 2 and refrigerated overnight. Rewarm in a skillet before serving.