In a large skillet, heat one tablespoon of olive oil over medium-low heat. Add the onions and season with one teaspoon of salt. Cook covered, stirring occasionally until softened and caramelized about 45–60 minutes. If the onions stick or begin to brown, add water one tablespoon at a time. Transfer onions to a large bowl and let cool.
While the onions are cooking, make the cream sauce. In a medium-sized saucepan, heat a drizzle of olive oil. Add the garlic, coconut milk, vegetable stock, nutritional yeast, and thyme and bring to a simmer over medium heat. Add the arrowroot flour, reduce heat to low and let the cream mixture simmer for 30 minutes. Set aside.
Prep the turnips by removing the greens. Wash and chop the greens. Peel and cut the turnip roots into ½ inch pieces.
To cook the turnips and greens, bring a large pot of well-salted water to a boil. Prepare a large bowl of ice water and keep it close by.
First, blanch the turnip roots by adding them to the pot of water and cooking for about 2 minutes. Using a slotted spoon remove the turnips from the water and immediately plunge in the ice water. Drain on a paper towel lined dish.
Next, cook the turnip greens in the boiling water but only for about 1 minute or until vibrant green and tender. Plunge into the ice water and then drain. Squeeze out excess water.
Add the turnips and greens to the bowl with the caramelized onions. Add the bread, cream mixture, vegan cheese, and season with salt and black pepper.
Preheat the oven to 375° F.
Grease a 13 x 9 inch baking dish with a drizzle of olive oil. Transfer the turnip/onion mixture to the baking dish. Using a spoon or your hands, press the ingredients down into an even compact layer. Top with more vegan cheese.
Bake the gratin, uncovered, until well browned, 50–60 minutes.
Let rest 10 minutes before serving.