In a medium saucepan, combine lentils, broth, water, and thyme sprigs. Bring to a boil, reduce heat to medium-low, and simmer, uncovered, until lentils are tender, about 20 minutes. Drain and set aside in a bowl to cool.
While lentils are cooking, roast red peppers over a gas flame or under the broiler, turning occasionally, until blackened. Place in a heat-proof bowl, cover with plastic wrap, and let stand until cool. Peel and seed peppers, and cut into 1/2 inch slices. Place strips in a small saucepan with 1/2 tablespoon oil; cover, and cook gently over low heat, stirring occasionally, until meltingly tender, about 10 minutes. Set aside, and let cool.
In a large bowl, whisk together vinegars, salt, thyme leaves, and remaining 2 tablespoons oil until blended. Add half of dressing to lentils, and toss to coat. Add arugula, chicken, and cooled pepper strips to remaining dressing, and toss. Season both salads with salt and pepper. Arrange lentils into mounds on each of six salad plates, form a well in the center of each, and fill with arugula mixture.