In a mini food processor, combine Parmesan, Greek yogurt, ¼ cup lemon juice, anchovy paste (or anchovies), Worcestershire sauce, and garlic and pulse.
With the motor running, add 2 tablespoons olive oil and puree until smooth.
Season with salt and pepper.
Heat a grill or grill pan over high heat.
Season chicken with salt and pepper and rub with olive oil.
Grill until golden and cooked through, 7 to 8 per side, then pour remaining lemon juice over chicken breasts and transfer to a cutting board.
Let rest 5, then slice into thin strips.
In a large bowl, toss kale and tomatoes.
Add half the Caesar dressing and taste, adding more if desired.
Divide salad among 4 plates and top with grilled chicken and more Parm.