To make the pizza by hand, combine the water and sugar in a bowl and stir in the yeast. Let stand for 10 minutes until foamy.
In a large bowl, combine the flour, salt, chile and pepper. Make a well in the flour and pour in the yeast water and olive oil. Stir until almost mixed, turn onto a floured board and knead until the dough is smooth and elastic.
Place the dough in a lightly oiled bowl and cover with plastic wrap. Place in a warm, draft free location and let rise until doubled, about an hour and a half.
Punch down the dough and divide into 2 balls. If preparing ahead of time, place in the refrigerator until ready to use. Bring the dough back to room temperature and then proceed with the recipe.
Roll out each portion into a round or oval pizza or do it free-form. If it will fit on your grill, you can also combine the balls into one and make one large pizza.
Heat a gas grill to hot. If using charcoal bank the coals to one side, creating a hot side and a warm side.
Brush each of the pizzas with olive oil and gently drape, oil side down on a hot grill. Shortly, within a minute or two, the dough will start to rise and bubbles will appear. Gently lift an end to see that the underside is browned and has grill marks. Immediately invert the crust onto a pan, and turn the gas grill to low.
Brush the dough with additional oil.
To make the Southwestern topping, lay the green chile strips over the cooked side. Top with the cheese and sprinkling of the olive oil. Slide the pizza(s) back onto the grill. Cover and cook, rotating once or twice until the toppings are heated through and the crust is browned, about 5 minutes on the cooler part of the grill.
