I start by separating the chicken quarters into drumsticks and thighs with one simple knife cut through the joint. This helps them cook more evenly and fit better in the tray. I peel and quarter the potatoes into even, chunky pieces so they cook at the same rate as the chicken.
In a large bowl, I whisk together the olive oil, lemon juice, garlic, mustard, oregano, thyme, salt, and pepper, leaving the chicken broth out for now. I add the chicken and potatoes to the bowl and toss everything until well coated in the marinade.
I spread the chicken and potatoes into a single layer in the roasting tray, placing the chicken skin-side up. I stretch the skin gently to cover as much of the meat as possible, since that's what gives you crispy, golden skin later on.
Now I carefully pour the chicken broth over the potatoes, not directly onto the chicken, so I don't wash off the marinade. I lay a piece of wet parchment paper over the tray, then seal it tightly with foil to trap steam.
I roast the tray at 425°F for about 45 minutes. This covered phase builds steam, keeps the chicken moist, and gives the potatoes a head start on becoming tender before the tray is uncovered.
Next, I remove the foil and parchment and roast uncovered for another 20 to 25 minutes. This is when the chicken and potatoes start to develop that gorgeous golden-brown color and the pan juices begin to concentrate.
For the final 4 to 5 minutes, I switch the oven to broil so the chicken skin crisps up and the potatoes get slightly charred edges. I check the internal temperature of the chicken with a thermometer to confirm it has reached 165°F.
While the tray rests, I quickly toss the shredded cabbage, dill, green onions, feta, oregano, olive oil, vinegar, salt, and pepper together in a bowl. I like dressing it right before serving so it stays crisp.