Arrange arugula, pickled onions, almonds, and Parmesan on a large platter.

Whisk together vinegar, olive oil, anchovy paste, Dijon, and honey; season with ¼ teaspoon each kosher salt and black pepper. Set aside. 

Cut each chicken breast in half crosswise to create two halves, one that’s naturally thicker and one that’s naturally thinner. Slice thick halves in half horizontally to create two thinner halves.

Arrange chicken pieces on a cutting board and cover with a piece of plastic wrap. Using a meat pounder (or rubber mallet) pound chicken pieces into an even ¼-inch thickness. 

Heat 2 teaspoons grapeseed oil in a large 12-inch skillet over medium-high until shimmering and just starting to smoke. Season both sides of chicken with kosher salt (1–1½ teaspoons total). 

Add half of the chicken pieces in an even layer and cook until bottom sides are golden brown, about 3 minutes. Flip and continue to cook until second side is golden brown and an instant-read thermometer inserted in the centers registers 163–165ºF, 3–4 minutes more. 

Transfer chicken pieces to a plate; season with pepper and set aside. Repeat with remaining 1 teaspoon grapeseed oil and chicken pieces.

Remove pan from heat, add shallot and cook until starting to brown, 2 minutes. 

Return pan to low heat and whisk in butter 1 tablespoon at a time (this is called mounting), scraping up any brown bits until shallots are golden brown, about 2 minutes. Pour in any released chicken juices (from plate of chicken) and cook, whisking constantly, until sauce is deep brown and thickened, 2–3 minutes.

Dip each chicken piece in pan sauce to coat then arrange next to salad on platter. Using a microplane, grate lemon zest over salad then drizzle with dressing; serve immediately.