Whisk vinegar and mustard together in a mixing bowl. Slowly drizzle in olive oil while whisking rapidly until the dressing is thick and creamy.
Hold an asparagus stalk at the stem, tip pointing away from you, on a work surface. Shave ribbons of asparagus off of the stalk with a vegetable peeler, moving the peeler from the stem-end to the tip. Repeat with remaining asparagus. Transfer asparagus strips to a bowl.
Sprinkle salt, black pepper, and cayenne pepper over asparagus; drizzle dressing over the top and toss to coat. Let sit until asparagus wilts slightly, 2 to 4 minutes.
Heat vegetable oil in a skillet over medium-high heat. Cook and stir pastrami in hot oil until almost crisp, 2 to 3 minutes.
Divide asparagus salad between four plates and top with pastrami.
