To make the custard, whisk together the half and half, sugar, flour, egg yolks, and salt in a medium pot, and then heat on medium-low heat, continuously stirring. After about 5-7 minutes, it will start to bubble and then thicken. Turn the heat down to low and stir in the butter and vanilla extract. Turn off the heat.
To make the whipped cream, take the bowl out of the freezer, add the cream and sugar, and then beat the ingredients until soft peaks form.
Peel and slice the bananas into 1/4” rounds, and then sprinkle the slices with the lemon juice to prevent them from turning brown.
To assemble the banana pudding, place a layer of 25 cookies and half the sliced bananas in the bottom of an 8-inch square baking dish. Evenly spread on top half of the custard. Add another layer of 25 cookies and the rest of the bananas, then top with the remaining custard.
Evenly spread on top of the pudding the whipped cream. Crumble the remaining cookies then sprinkle the crumbs on top of the whipped cream.
Alternatively, you can serve the pudding in 8 1/2-pint Mason jars or bowls. Evenly divide the cookies, sliced bananas, and pudding, and then proceed as above.
Chill covered for at least 4 hours before serving, though if you prefer, you can eat it warm. It will keep refrigerated for 3 days.