Step 1In a small bowl, stir together the yogurt, 2 tablespoons of the olive oil, the dry mustard, thyme, coriander, 1 1/2 tablespoons salt and one-half teaspoon pepper. Loosen the skin around the chicken breasts and thighs, then rub both chickens all over (beneath the skin and inside the cavity, too) with the yogurt mixture. Refrigerate the chickens, uncovered, for 1 hour.
Step 2Heat the oven to 350 degrees. Place the shallots, carrots, peppers, the remaining 2 tablespoons of olive oil, three-fourths teaspoon salt and one-fourth teaspoon pepper into a large roasting pan and toss well. Arrange a rack over the vegetables large enough to hold both chickens.
Step 3Arrange the chickens on the rack, breast-sides up, and roast, basting occasionally with pan juices, until the vegetables are very tender and the chickens are deep golden brown and cooked through, 1 1/2 to 2 hours. Transfer the chickens to a large platter and tent with foil; set aside.
Step 4Drain the pan drippings into a bowl, then skim off and discard the fat; set aside.
Step 5Remove and discard the skin from the peppers (it should peel off fairly easily), then transfer them to a food processor. Add half the shallots and pulse until roughly chopped. Add the goat cheese and salt and pepper to taste and pulse again until just combined. This makes a generous 2 cups of puree. Remove all but 1 cup of the puree and set aside to use throughout the week as a sandwich spread or dip, then puree the remaining red pepper mixture until smooth.
Step 6In a large skillet, whisk together the reserved pan drippings, chicken broth and flour until smooth, then bring to a boil over medium-high heat. Reduce the heat to medium-low and simmer until just thickened, about 1 minute. Whisk in the red pepper puree and cook until just thickened, 1 to 2 minutes more. Strain the sauce, discarding any solids, and keep warm.
Step 7Carve the chickens and transfer to plates. Spoon 1 to 2 tablespoons of the red pepper and goat cheese sauce over each serving and serve with the remaining roasted shallots and carrots on the side.