To make the dressing. Combine all ingredients in a glass jar and whisk until smooth. Taste and adjust the salt and pepper.
In a bowl, toss the chicken with 1/3 of the dressing. Let sit 10 minutes or marinate up to overnight. Set your grill, grill pan, or skillet to medium-high heat. Grill the chicken until lightly charred and cooked through, turning halfway through cooking, about 8-10 minutes.
Bring a large pot of salted water to a boil. Boil the pasta to al dente, according to package directions. Drain. Add the pasta, chicken, mozzarella, tomatoes, and avocado to a large salad bowl. Pour over the remaining dressing and toss well to combine. Serve warm or cold. The salad will develop more flavor as it sits.