In a large pot, bring water to boil and cook pasta to al dente. Drain, reserving 1/2 cup pasta water for later.
While pasta is cooking, add olive oil to large skillet. Heat over medium-high and add capers. Cook, stirring frequently, until crispy. Remove capers with slotted spoon to plate.
Add chicken to same skillet and season with salt and pepper. Cook, 6-8 minutes per side, until chicken is done. Remove chicken to plate and cut into bite-sized chunks.
In same skillet, add reserved pasta water, white wine, lemon juice, Parmesan cheese, garlic, and thyme. Bring to boil, then reduce heat to simmer for 5 minutes.
Stir in chicken and capers to sauce, then remove from heat. Toss sauce/chicken mixture with cooked pasta and serve immediately. Garnish with fresh grated Parmesan cheese and extra fresh thyme, if desired.