Place oil and butter in a large skillet over medium heat. Melt butter and mix in potatoes, parsnips, carrot and shallot. Season with salt and pepper. Cook 20 minutes, stirring occasionally, until potatoes are almost tender.
Add the squash, mushrooms and kale. Continue cooking for 5-8 more minutes, until the squash is tender. You may need to add an extra splash of oil if the mixture is drying out. Season to taste with salt and pepper. Garnish with fresh thyme leaves.