Place pork pieces between two sheets of heavy plastic (resealable freezer bags work well) on a solid, level surface. Firmly pound pork with the smooth side of a meat mallet to a thickness of ½ inch.

Transfer pork pieces to a plate and season both sides generously with salt and black pepper; sprinkle with 2 tablespoons of flour, lightly coating both sides. Pour eggs over pork, turning to coat.

Place panko in a shallow bowl; place pork pieces in panko, pressing firmly to coat on both sides. Transfer breaded cutlets to a clean plate, cover with plastic wrap, and refrigerate for 15 minutes.

Meanwhile, melt butter in a skillet over medium heat. Stir in pickles, jalapeño pepper, and green onions; cook and stir until onions have softened, about 3 minutes. Sprinkle in 1 ½ tablespoons flour; cook and stir for 3 minutes.

Slowly stir in 2 to 3 tablespoons cold milk, whisking constantly. Stir in remaining milk, Worcestershire sauce, and black pepper. Simmer on low for 5 minutes. Taste and adjust seasoning; set sauce aside.

Heat ¼ cup vegetable oil in a large skillet over medium-high heat. Add ½ of the breaded pork cutlets and cook until pork is no longer pink inside, and the crust is well-browned, 4 to 5 minutes per side.

Transfer to plate lined with paper towels. Sprinkle with salt. Repeat with remaining vegetable oil and pork cutlets.

Serve cutlets topped with sauce.