I have the same book I believe. Chef McPeake's Book from Kansas City?
Asian chile paste (p. 45) refers to a general term covering many different chile sauces, variously labeled as chile paste, chile sauce, and chile paste with garlic. Best brand is Rooster Brand Delicious Hot Chile <garlic> Sauce. I believe they make one also without the garlic in it which is what I just ran out of. On page 79, the Asian Chili Garlic Paste would refer to the same with the garlic in it. You can find it sometimes in most grocery stores, generally they would carry Lee Kum Kee brand, or Dynasty Brand. Best to get this at your local Asian grocery store.
On edit: I do believe that the 'asian paste' is probably an error due to proofreading. There are a few in the book. Another thing I noticed when I went thru it, instructions for how to place your meat, fat up or down, or distinguishing different smokers is not talked about. His recipes though, are fabulous.