If desired, peel the carrots and parsnips. Cut in half lengthwise and then into ~1/4" thick pieces. Peel the rutabaga and cut into similar-sized pieces to the carrot and parsnip. If you make the pieces too thin, you'll end up with a pot of mush.
In a large, heavy bottomed pot, like a Dutch oven, heat the ghee or coconut oil over medium-high heat. Toss in the carrots, parsnips, rutabaga and onion. Allow the veggies to caramelize and slightly brown, though not burn. Stir enough so you get good colour on many of the pieces.
Add the garlic, rosemary, thyme, sage and chicken broth. Allow the liquid to come to a boil, then reduce the heat to low. Cover the pot. Allow the veggies to cook until just fork-tender, about 20 minutes.
Uncover the pot. Turn the heat to high and allow the braising liquid to reduce down until it's a slightly thickened sauce. Adjust seasoning with salt and pepper if desired.