First, you make your dough (if you're going that route).  The one I use is super easy and super delicious.  Or you can definitely pick up some from the store.

Then make the simple no-cook tomato sauce by blitzing some high-quality canned tomatoes (I love San Marzano) and a few other ingredients. 

So I have like 5 different methods for awesome grilled pizza, but here is my current favorite. Lay a piece of parchment paper on a cutting board or platter.  Roll or stretch your dough out quite thin, and into a circle...or quite frankly whatever the hell shape it wants to be.  South America is a(n accidental) favorite of mine.  

Go ahead and top your pizza.  A *thin* layer of the tomato sauce, then some nicely-spaced torn fresh mozzarella or just the little pearls.  You definitely want to use the nice stuff and not regular shredded mozzarella.  Sprinkle some fresh-grated parmesan on there too.

Then use scissors to cut around the parchment very close to the dough.  I had issues with all the excess parchment ending up as black burnt bits in my pizza---this method eliminates that problem.

Preheat your grill to medium, probably around 500 degrees.  When it's hot, gently slide the pizza and parchment paper onto the grates. 

Close the lid and let cook for 2-3 minutes.  Then (this will depend on your grill, so you may need to play around a bit) you can use tongs to slide the parchment paper out and discard it.

For me, I've found I need to then either turn the heat directly under the pizza down significantly, or off (and keep the other burners on for more indirect heat).  On my old grill I could keep the burners as-is, so it will depend on how intense your grill is (my new Weber's pretty intense).

You'll want to keep a very close eye on the bottom, you don't want to burn it.  Once your cheese is melted and the dough seems cooked through and crisp, use tongs, a pizza peel, or a large spatula to slide it off the grill and onto a large platter or cutting board.

Let it sit for five or so minutes, then top with fresh basil.  Cut and serve immediately.