Bring the turkey breast to room temperature: set it on a rimmed baking sheet or in a roasting pan and let it sit for 45 to 60 minutes. Pat it very dry inside and out with paper towels — dry skin is what turns crisp.
Preheat the oven to 425°F (220°C) with a rack in the lower third. Scatter the onion quarters in a roasting pan fitted with a rack and pour in the broth.
Make the herb butter: in a small bowl, mash the softened butter with the garlic, sage, rosemary, thyme, lemon zest, salt, and pepper until evenly mixed and fragrant.
Butter under the skin: gently work your fingers under the skin over the breast, loosening it without tearing. Spread about two-thirds of the herb butter directly onto the meat under the skin and massage it through the skin to distribute it evenly. Rub the remaining butter all over the outside of the skin.
Season and stuff: rub the outside with the olive oil, salt, and pepper. Tuck the lemon halves, halved garlic head, thyme, and rosemary sprigs into the cavity under the breast bone or around the pan.
Roast hot, then lower: roast at 425°F for 30 minutes, until the skin starts to turn golden. Reduce the heat to 325°F (165°C) and continue roasting for 1 hour 15 minutes to 1 hour 30 minutes, basting with the pan juices every 30 minutes. If the skin browns too quickly, tent it loosely with foil.
Check the temperature: the breast is done when an instant-read thermometer inserted into the thickest part (without touching bone) reads 160°F (71°C). The temperature will rise to 165°F as it rests.
Rest before carving: transfer the breast to a cutting board, tent loosely with foil, and rest for 20 to 25 minutes. Do not skip this — the juices redistribute and the meat stays juicy. Save the pan drippings for gravy, then slice against the grain and serve.