In a medium bowl, mix together 1 1/2 cup flour and beer with a wooden spoon; let stand for at least 3 hours at room temperature.
Mix eggs and milk in a small bowl. In a separate bowl mix together 1/2 cup flour and salt and pepper.
Heat oil to 375 degrees F (190 degrees C).
Working in batches, dip each vegetable into egg mixture, seasoned flour, and finally in beer batter. Place vegetables into oil and fry until golden brown. Transfer to a paper towel-lined plate to absorb excess oil. Serve immediately.