For the chicken: Preheat the oven to 400 degrees F.
Dry off the chicken well with paper towels. Put it in a roasting pan and massage with some of the oil, some of the butter and 2 tablespoons salt.
Add the onion halves to the pan, drizzle in the remaining oil and sprinkle with some salt. Add the remaining butter to the pan. Squeeze the juice from the lemon halves over the chicken and into the pan, then place the lemon halves in the cavity of the chicken along with the thyme sprigs. Add the white wine to the pan.
Put the chicken in the oven, reduce the temperature to 350 degrees F and roast, basting halfway with the pan juices, until the thickest part of the chicken measures 160 degrees F, about 1 hour. Allow to rest 20 minutes before carving.
For the shallot fried rice: Put the rice in a medium saucepan with 2 cups water and bring to a simmer over medium-high heat. Cover the pan, reduce the heat to low and cook until the water is absorbed and the rice is tender, about 20 minutes. Let stand, covered, for 10 minutes.
Meanwhile, heat some oil in a saute pan over medium heat until it begins to shimmer. Add the shallots and cook, stirring, until browned, about 7 minutes. Transfer the shallots to a plate or bowl using a slotted spoon and sprinkle with a generous amount of salt (use a lot because the shallots will be the "salt" for your rice). Add the mushrooms to the remaining oil in the pan and cook, stirring occasionally, until browned.
Mix together the rice, mushrooms and shallots in a serving bowl. Serve the chicken tossed with the arugula along with the shallot fried rice.
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