Heat the oven 400°F. Spray a baking sheet with nonstick cooking spray or line with a nonstick silicone baking mat.
Heat oil in a large saute pan over medium heat. Add the onion and cook until clear and translucent, about 3 minutes. Add the garlic and cook until fragrant, 45 to 60 seconds. Transfer the vegetables to a large bowl to cool slightly. (You can also just zap it in a bowl in the microwave, but let it cool a bit before the next step.)
Add the turkey and egg. Season with salt and pepper and mix well with your hands. (So that you can taste and adjust for seasoning, simply zap a teaspoon or so in the microwave. Add salt and pepper, as needed.) Shape into one large patty about 1 inch thick. Place on one side of the prepared baking sheet. Combine the ketchup and Worcestershire; brush the meat with the mixture.
Place the broccoli in a bowl. Spritz with oil. Season with salt and pepper. Toss to coat. Place the broccoli on the baking sheet opposite the turkey. Transfer to the oven and cook until the turkey meatloaf registers 165°F when measured with an instant read thermometer, about 30 minutes. Serve immediately.