A WHIP hat, so called from the want of depth in the crown. LILLY SHALLOW, a WHITE Whip hat.
Definition taken from The 1811 Dictionary of the Vulgar Tongue, originally by Francis Grose.
Shallow Pate * ShamNathan Bailey's 1736 Dictionary of canting and thieving slang
John S. Farmer's collection of canting songs and slang rhymes