Make the blackening rub by combining paprika, garlic powder, onion powder, dried thyme, cayenne, salt, and black pepper in a small bowl. Pat chicken dry and rub both sides with olive oil, then press the spice mix over the breasts so they’re evenly coated.
Heat a heavy skillet (cast-iron preferred) over medium-high heat until very hot. Add a thin film of oil. Sear the chicken 3 to 4 minutes without moving it so a dark crust forms, then flip and cook another 3 to 4 minutes. Lower heat if the spices begin to burn. Cook until internal temperature reaches 160–165°F. Transfer chicken to a plate and let rest 5 minutes before slicing thinly.
While the chicken cooks, make the dressing: mash the anchovy paste (or minced fillets) with minced garlic and a pinch of salt in a bowl. Stir in mayonnaise, lemon juice and lemon zest, Dijon, then slowly whisk in the olive oil until emulsified. Stir in grated Parmesan and taste; adjust seasoning with salt and pepper.
Prepare the quick croutons: warm a separate skillet over medium heat. Add oil or butter and a smashed garlic clove to perfume the fat, then toss in bread cubes and cook, stirring often, until golden and crisp, about 6–8 minutes. Season lightly with salt and discard the garlic clove.
Assemble the salad: place torn romaine in a large bowl, add about half the dressing and toss to coat (add more if you like it saucy). Divide the dressed lettuce among plates or bowls, top with sliced blackened chicken, sprinkle with grated and shaved Parmesan, scatter warm croutons, and finish with a lemon wedge.
Serve immediately so croutons stay crisp and chicken remains juicy. If preparing ahead, keep dressing and croutons separate until serving.