Heat a grill pan over medium-high heat until very hot, about 4 minutes. On a lightly floured surface, roll dough into a 7 by 14-inch oval.
Lightly brush both sides of the dough with olive oil and place on hot grill. Grill until lightly charred and the dough begins to puff up, 2 to 4 minutes. Flip dough and grill on the second side until charred and just cooked, 2 minutes. Remove pizza from grill and set aside.
In a medium mixing bowl, combine the tomatoes, basil, artichokes, and just enough of the artichoke marinade to coat the mixture. Toss gently and scatter over the pizza. Top with basil, Parmesan, and season with salt and pepper, to taste. Serve immediately.
In a small mixing bowl, whisk together yeast, warm water, sugar and olive oil. Let sit for about 3 minutes to fully dissolve and activate the yeast.
In a large mixing bowl, whisk together flour and salt.
With a rubber spatula, gradually mix the yeast mixture into the flour until just combined and dough barely holds together. Turn the dough out onto a clean counter surface dusted with flour.
Knead the dough with flour-dusted hands until the dough has become smooth and elastic, about 10 minutes. The surface should be tight and silky and bounce back slightly when pressed. Lightly grease a large mixing bowl with olive oil and place the dough in it.
Cover the bowl tightly with plastic wrap and set in a warm place to rise until doubled in size, 1 to 2 hours.
Once doubled, punch the dough down and divide it into 2 equal pieces. Knead each 1 slightly to form a uniform ball. Set aside and cover them with a kitchen towel or plastic wrap to rest for 30 minutes.
If grilling outdoors, roll pizza into a circle or oval about 1/3-inch thick. Dough can be made the day before and refrigerated overnight once it is divided. Keep covered tightly in plastic wrap or in a sealed food storage bag.