Prepare Pasta Water: Heat a large pot of salted water to boil while you cook the chicken.
Heat a large skillet to medium high heat. Season both sides of chicken thighs with salt and pepper. Add olive oil to the skillet and cook chicken 5-6 minutes each side until 165°F internally.
When cooked, set chicken aside to cool and keep skillet on medium heat.
Once the hot water is boiling in the pot, add the pasta and olive oil, cook until al dente (tender, but firm and chewy), about 11-13 minutes. The olive oil will help the pasta to not stick together. Stir every few minutes to keep pasta from sticking to the bottom of the pot.
Add the white wine to the skillet, then add the kale and tomatoes. Roast about 3-4 minutes until tomatoes are blistered and kale is softened. Once cooked, scrape the pan and set vegetables aside. Keep the skillet on low heat.
Add butter and garlic to the skillet and cook 1-2 minutes until the butter is melted and garlic is fragrant. Add the heavy cream, sherry vinegar and capers, whisking for one minute and increase the heat to medium. Add the cheese and continuously stir until thickened and the cheese is melted, about 5-8 minutes.
Remove and strain the pasta and set aside for serving. You can add a dash of olive oil to the pasta to keep it from sticking while you finish preparing the dish.