Preheat the oven to 450 degrees Fahrenheit.
Grease the interior bottom and sides of a large 4.5 quart baking dish (usually a Pyrex or glazed ceramic dish) with olive oil or another approved cooking fat.
While the oven preheats, remove the chicken from its packaging, remove the giblets from the interior cavity and save for another use or discard. Pat the chicken dry with paper towels, then place it breast-up in the greased baking dish.
Carefully slide your fingers underneath the skin on the breasts (starting from the bottom near the cavity), creating a pocket between the skin and actual chicken breast flesh. Wash your hands well with hot water and soap before moving on to the next step.
Next, sprinkle the outside of the chicken well with kosher salt (about 1 tablespoon total) and ground black pepper.
In a small bowl, stir together the paprika, cumin, cayenne, and lemon zest. Spoon out 1 tablespoon of the spice rub mixture for the vegetables and set aside.
Add two tablespoons of olive oil to the main spice rub mixture (NOT the 1 tablespoon you reserved) to create a paste, and rub it all over the chicken. Start with the pockets between the skin and chicken breasts, and then rub the rest over the entire outside of the chicken. If you have any left, you can rub it inside the chicken cavity too.
Finally, stuff the lemon quarters inside the chicken cavity, and place the whole baking dish in the oven to roast, uncovered, for 30 minutes.
While the chicken roasts, place the chopped sweet potatoes, cauliflower, and onion in a large bowl. Drizzle with 3 tablespoons of olive oil, and sprinkle generously with kosher salt and pepper. Add the reserved spice mixture to the vegetables, and toss until evenly coated.
When the chicken has roasted for 30 minutes, remove the baking dish from the oven and place on a stove-top or heat-proof surface. Carefully spoon the vegetables into the pan, surrounding the chicken entirely. Be careful not to accidentally touch the hot baking dish!
When the vegetables are evenly nestled around the chicken, place the whole baking dish back in the oven, reduce the heat to 425 degrees Fahrenheit, and let cook for another 45 minutes.
When the 45 minutes are up, use a meat thermometer to verify that the thickest part of the thigh has reached at least 165 degrees Fahrenheit. If it has, then the chicken is done. If the chicken has not reached the proper temperature yet, continue roasting until it does.
When the chicken is done, remove it from the oven and let rest for 10 minutes before slicing and serving alongside the roasted vegetables and pan juices.