Place chicken in a skillet or saucepan and add enough water to cover; bring to a boil. Cover, reduce heat to low and simmer gently until cooked through and no longer pink in the middle, 10 to 12 minutes. Transfer the chicken to a cutting board to cool. Shred into bite-size strips.
Bring a large pot of water to a boil. Cook pasta until just tender, about 9 minutes or according to package directions. Drain. Place in a large bowl.
Meanwhile, place oil and shallot in a small skillet and cook over medium-low heat, stirring occasionally, until softened and just beginning to brown, 2 to 5 minutes. Add broth, feta and lime juice and cook, stirring occasionally, until the feta begins to melt, 1 to 2 minutes.
Add the chicken to the bowl with the pasta. Add the dressing, blueberries, thyme, lime zest and salt and toss until combined.