Put the chicken in a wide pot. Add the broth, olive oil, fennel fronds, thyme, garlic, 1 1/2 teaspoons salt and 1/2 teaspoon pepper, then add enough water to cover the chicken and turn the heat to medium low. Poach the chicken, turning occasionally and keeping the liquid just below a simmer, until just cooked through, about 30 minutes (a thermometer inserted into the center of a chicken breast should register 155 degrees F to 160 degrees F). Remove the chicken to a plate with a slotted spoon and let cool to room temperature. Increase the heat to medium high and simmer the cooking liquid until the garlic is tender, about 5 more minutes; remove the garlic and set aside. Strain the cooking liquid; reserve 1/4 cup and set aside. While the chicken is cooking, place the red peppers directly on a stovetop gas burner over high heat and char on all sides, turning with tongs, about 10 minutes. (If you don't have a gas stove, char the peppers on a baking sheet under the broiler, turning occasionally, about 12 minutes.) Transfer to a bowl, cover with plastic wrap and set aside, 20 minutes. Peel the skin off the peppers with your fingers; discard the stems and seeds and chop.
Make the aioli: Puree about three-quarters of the roasted red peppers in a blender with the mayonnaise, reserved garlic and 1 tablespoon lemon juice until smooth. Season with salt and pepper. Make the salad: Thinly slice the fennel bulbs (use a mandoline if you have one) and transfer to a large bowl. Finely chop the remaining roasted red peppers, then add to the bowl along with the parsley, the remaining 2 tablespoons lemon juice and the reserved 1/4 cup cooking liquid; toss. Season with salt and pepper. Slice the chicken; top with the aioli and serve with the fennel salad. Photograph by Anna Williams