Place pork 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 1/2 inch.
Place pork pieces on a plate and season both sides generously with salt and black pepper. Sprinkle 2 tablespoons of flour over pork pieces, lightly coating both sides. Pour eggs over pork, turning to coat.
Transfer pork pieces to a bowl filled with panko bread crumbs. Press bread crumbs firmly into the meat on both sides. Transfer breaded cutlets to a clean plate, cover with plastic and refrigerate for 15 minutes.
Melt butter in a skillet over medium heat. Stir in pickles, jalapeno pepper, and green onions; cook and stir until onions have softened, about 3 minutes.
Sprinkle in 1 1/2 tablespoon 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.
Heat half the vegetable oil in a large skillet over medium-high heat. Add 4 breaded pork cutlets and cook until pork is not 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.