Preheat the oven to 400°F. Toss the bread cubes with 3 tablespoons of the olive oil and 1 teaspoon of the kosher salt, and spread evenly on a rimmed baking sheet. Bake in the preheated oven until dry and just beginning to brown, about 15 minutes.
Toss together the squash, beets, turnips, carrots, pepper, and remaining 1⁄4 cup olive oil and 1 1⁄2 teaspoons salt; divide between 2 rimmed baking sheets, and spread in an even layer. Bake at 400°F until tender and lightly browned, about 30 minutes, stirring twice during baking. Cool completely on baking sheets on wire racks, about 20 minutes.
Toss together the bread cubes, roasted vegetables, and Vinaigrette in a large bowl until combined. Let stand 10 minutes.
Place the arugula on a platter; top with the roasted vegetable mixture, and sprinkle with the goat cheese. Serve immediately.