Which brewing method extracts more flavors from the coffee bean?
More is not always better. To extract the most flavour, just let it simmer for a long time... it just won't be a nice cup of coffee, because that way some unwanted flavours are also extracted.

To make a good cup of coffee, the water, the coffee, the temperature (also of the cup itself), the time, the grind and the blend all are important. Just as important as the method.

Favourite methods are the French press and the espresso machine, but a radical different method is Turkish' coffee, where they take the coffee to a boil three times and serve it without filtering... and it's delicious.