Whisk together mayonnaise, olive oil, dijon mustard, lemon juice, Worcestershire, and garlic until smooth.
Whisk in parmesan and season with salt and pepper. Refrigerate until ready to use.
Preheat oven to 350F and line a baking sheet with parchment paper.
Toss sourdough cubes with olive oil, salt, and garlic powder.
Spread on baking sheet and bake 15 minutes, tossing halfway, until golden brown.
Pound chicken breasts to 1/4-inch thickness between plastic wrap.
Season both sides with salt and pepper.
Set up breading station with flour mixed with salt and pepper, beaten eggs, and breadcrumbs mixed with parmesan.
Dredge each chicken breast in flour, dip in eggs, then press into breadcrumb mixture.
Heat 1 Tbsp butter and 1 Tbsp olive oil in a large skillet over medium-high heat.
Add 2-3 chicken breasts and sear 4-5 minutes per side until golden and internal temp reaches 165F.
Transfer to cooling rack and sprinkle with flaky sea salt. Repeat with remaining chicken.
Toss chopped lettuce with Caesar dressing until lightly coated.
Add croutons and grated parmesan, tossing gently.
Slice chicken and arrange over salads.
Top with shaved parmesan, black pepper, and lemon wedges.