It is a normal cap shaped like a Krabby Patty. There are buns with sesame seeds both on top and the bottom of the hat. A meat patty lies on the center with some lettuce.
Role in series
"Mutiny on the Krusty"
Thaddeus is seen wearing the hat and claims to wear it because he is bald. When Mr. Krabs flies out of the Krusty Krab, Thaddeus bows his head and then takes off his hat to take a bite. The Hatty Patty was also the idea of Squidward, and Mr. Krabs makes it.
"Frozen Krabby Patty Song"
One of the types of Krabby Patties that Patrick shows off is a Hatty Patty. He takes it off of his head, shows his bald head, and takes a bite out of the hat, revealing it to be edible.