Make the chicken: Preheat the oven to 425 degrees F. Peel 2 garlic cloves and transfer to a mini food processor; set the rest of the head of garlic aside. Grate the blood orange zest into the food processor (set the oranges aside for the salad). Pulse until the garlic is finely chopped. Add the parsley, sage, red pepper flakes, 1 teaspoon salt and a few grinds of pepper; pulse, scraping down the food processor as needed, until the herbs are finely chopped. Add the butter and pulse to combine.
Pat the chicken dry with paper towels. Gently slide your fingers under the skin on the breasts and legs to loosen it. Spread the orange-herb butter under the skin. Season the inside of the chicken with salt and pepper, then stuff the rest of the garlic head into the cavity. Tuck the wings under the chicken and tie the legs together with twine.
Set a rack in a large roasting pan; put the chicken on the rack and rub with olive oil. Season generously with salt and pepper. Roast 30 minutes, then rotate the pan and reduce the oven temperature to 350 degrees F. Continue roasting until the skin is golden brown and a thermometer inserted into the thigh registers 165 degrees F, about 1 hour. Let rest in the pan 15 minutes.
Meanwhile, make the salad: Fill a large bowl with ice water. Thinly slice the fennel (use a mandoline if you have one) and add to the ice water; let sit 20 minutes. Combine the fennel fronds, beans, olive oil, 1/2 teaspoon salt and a few grinds of pepper in a large bowl. Grate the zest of 1 blood orange into the bowl; set aside.
Remove the peel and pith from all 6 blood oranges using a paring knife. Halve the oranges and thinly slice. Drain the fennel and dry in a salad spinner. Refrigerate the oranges and fennel until ready to use.
Transfer the chicken to a board. Add the fennel, blood oranges, olives, parsley and mint to the bowl with the beans. Season with salt and pepper and toss. Serve with the chicken.