Leave them in the ground longer so that they have a longer growing season. Also, make sure that they are well watered.

And some varieties of potato do not produce large potatoes for example the Pink Fir Apple.

Choice of variety is the main variable, so look for a 'late' potato and one which is recommended for making chips or baked potatoes.
Most of the earlies and mid season ones you see for sale tend to be salad potatoes which are quite small and recommended for quick growing in the spring. Delicious with butter, but I know what you mean about having some monsters which you can store and keep for winter.

My favourites are King Edward and Maris Piper (although M. Piper isn't a particularly large one) My dad always grew Ulster Prince which, if I remember from my childhood, are enormous. The only problem with them is finding a supplier!

Good luck.