Fill a 5-quart / 4.7 L pot with water, add the 1 tablespoon kosher salt, and bring it to a boil. Halve each chicken breast horizontally into two even cutlets about 1.5 cm / 5/8 inch thick, which gives you 6 cutlets. Pat them dry and season both sides with the garlic powder, 3/4 teaspoon salt and 1/2 teaspoon pepper.
In a 4-quart / 3.8 L bowl, mash the anchovies and garlic into a paste with a fork. Whisk in the Greek yogurt, 2 tablespoons olive oil, lemon juice, Dijon, Worcestershire, 30 g grated Parmesan and 1/2 teaspoon pepper until the dressing is thick and pale. Halve the tomatoes and chop the romaine now and keep them aside.
Add the rotini to the boiling water and cook 10 to 11 minutes until al dente. While it cooks, heat 1 tablespoon olive oil in a 30 cm / 12-inch skillet over medium-high heat until it shimmers. Sear 3 cutlets for 3 to 4 minutes per side until golden and 165 F / 74 C in the thickest part, then move them to a plate and repeat with the other 3.
Drain the pasta, rinse it under cold water for 20 seconds, and shake it dry. Fold it into the dressing until every piece is coated, adding 1 to 2 tablespoons of water if it looks stiff. Rest the chicken 5 minutes, then slice it across the grain into 1 cm / 1/2 inch strips and add it to the bowl.
Stir in the tomatoes, cover, and chill for 15 minutes so the pasta takes on the dressing. Right before serving, fold in the romaine, taste, and add lemon juice or salt if needed. Top with the 10 g shaved Parmesan and serve within 30 minutes.