I Don’t have a mixer with a paddle blade.

Yes. You can make this in a food processor but make sure the butter is very cold and only process until it is crumbly like cornmeal not until it forms into dough. Turn out on to counter or into a bowl and use you hand to bring crumb mixture into a ball.

You could use a food processor to make this, but it will change the texture slightly. If your mixer has the wire style beaters, or even the wider band sort, it will work out just fine to use that. A paddle is the ordinary mixer attachment for Kitchen Aid and other makes of stand mixers- but the regular sort provided with hand mixers works just fine. It might take a bit longer to work the butter and sugar in, and you may have to work the flour in by hand if you mixer doesn't have enough power for it. If you happen to like crumbly shortbread, the food processor will deliver that. A smoother texture will result from the mixer. Your pick on which you prefer.