There are two simple ways to dispatch your lobster kindly; if they die without stress the texture will be better.
The first method is to put the lobster in the freezer for an hour or two, which puts it to sleep, and is by far the easiest way to do it.
Or put the lobster in a sink filled with fresh water, putting a chopping board over the top as it will thrash about a bit. You are drowning the lobster.
Whichever way you have taken your lobster’s life, now put the lobster on a chopping board with its head facing you and its tail away. Carefully place a sharp knife between its eyes. Holding the tail, cut down towards you, right through the shell. Turn the lobster around and, holding it by the head, cut through the tail; you should now have two halves. Pull out the digestive tract that runs down its length.