Preheat the oven to 450 degrees F.
In a large (I used 13-inch) oven-proof skillet over medium heat, warm the olive oil and melt the butter. Place the cabbage wedges cut side down (on one of the cut sides), season with salt and pepper and cook until browned, about 10 minutes. (You can start the farro now - go to step 5)
Place the skillet with the cabbage in the oven and cook for 15 minutes. Flip the wedges to the other cut side down and cook for an additional 10 minutes.
If the cabbage doesn’t have enough color, turn the broiler on and broil until cabbage is lightly charred, approximately 1-3 minutes.
While the cabbage is cooking, prepare the mushroom farro.
Place all ingredients into a medium saucepan or stockpot in the order listed, with farro first, then onion, etc. Bring uncovered pan to a boil over high heat, lower heat to a simmer, partially cover and stir occasionally. After 30 minutes, farro should be tender. If it's not, cover and continue to simmer for a few more minutes. There should be a little liquid at the bottom of the pot, but it should not look like soup. Turn off the heat and cover until ready to serve. The longer it sits, the more liquid will be absorbed. Personally, I like this a little juicy.
Remove the sprig of thyme and discard it. Transfer the farro to a platter or shallow bowl and top with the cabbage wedges.
Garnish with parsley and/or chives.