Special equipment: a heavy-duty half-sheet pan, optional (see headnote)
Position 2 oven racks on the highest and lowest settings and place a heavy-duty rimmed half-sheet pan on the bottom rack. Preheat the oven to 450 degrees F.
Place the chicken thighs in a large bowl and toss with 2 tablespoons of the oregano, the zest of 1 lemon, 2 tablespoons oil, 1/2 teaspoon salt and some black pepper. Place the chicken skin-side up in the center of the heated baking sheet and return to the oven while you prepare the vegetables.
In a second bowl, toss the potatoes and carrots with 2 tablespoons of the oil, the lemon halves, the remaining lemon zest the remaining 2 tablespoons oregano,1 teaspoon salt and some pepper. Spread the vegetables and lemons (cut-side up) around the chicken on the sheet pan and roast on the bottom oven rack for 10 minutes. Move the sheet pan to the top rack and roast until the vegetables are tender and the chicken skin is crispy and the meat registers 165 degrees F on an instant-read thermometer, about 20 minutes more. Remove the chicken and vegetables to a serving platter and keep warm. Remove the lemon halves to a cutting board to cool for a minute.
Set the sheet pan on the stovetop over medium-high heat and bring the pan juices to a boil. Squeeze in the lemon juice, reduce the heat to a simmer and whisk in the remaining 1 tablespoon oil. Scrape up the crusty bits from the bottom of the sheet pan with a wooden spoon to incorporate into the sauce and season with salt and pepper to taste. Pour the juices over the vegetables, garnish with oregano sprigs and serve.