To do something to its complete extremity. To partake in something and put in more than the adequate amount of effort required that it becomes completely over the top and unnecessary.
Example #1. "I told my friend to make himself at home but he went big Asian style and ate all my food, redecorated my house, painted the walls and shat in my garden."
Example #2. Friend #1: "Dude, I'm going to get on the piss big Asian style tonight!"
Friend #2: "Well, I look forward to scraping your ass off of the curb after you've rooted the ugliest, fattest girl in the bar, made a complete ass out of yourself and streaked down the road before passing out in a pool of blood, vomit and urine..."