Rub the chicken all over, including the cavity with 1 teaspoon herbes de Provence, 1 tablespoon salt and 1/4 teaspoon pepper. Refrigerate for at least 2 hours or up to overnight; bring the chicken to room temperature 30 minutes before cooking.
Arrange the rack to the lower third of the oven and preheat to 425 degrees F.
Butter the sides and the bottom of a 10-cup Bundt pan. Cover the tube hole in the middle of the pan with a 6-inch piece of foil. Put the potatoes in the pan along with the 3 smashed garlic cloves, 1/2 teaspoon salt and a few grinds of pepper. Toss with 1 tablespoon melted butter. Roast for 20 minutes.
Meanwhile, mix together the lemon zest, paprika, remaining 4 tablespoons butter, 1 teaspoon herbes de Provence and grated garlic in a small bowl. Pat the chicken dry and spread a little of the butter mixture under the skin of the breasts and thighs. Rub the rest all over the chicken. Sprinkle with a little more salt and pepper and place the chicken cavity on the foil-covered tube over the foil so that the chicken is sitting upright on top of the potatoes.
Cook until an instant-read thermometer inserted in the thickest part of the thigh (avoiding the bone) reads at least 165 degrees F, 50 to 55 minutes. Transfer the chicken to a cutting board and let rest 10 minutes before carving.
Transfer the potatoes with a slotted spoon to a serving dish. Strain the liquid in the bottom of the pan into a small saucepan and heat until just warmed through. Squeeze in the lemon juice and stir in the parsley. Pour the cooking liquid over the carved chicken and serve with the potatoes on the side.
When using dried herbs, crush them in your hands before adding them to the other ingredients to bring out their flavor.
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