If guests are gathering near the grill during cocktail hour, have all toppings at the ready.
Cut dough into 2 pieces, and dust lightly with flour. Roll out one piece on a well-floured surface, forming a circle about 6 or 7 inches in diameter. (Leave crust a bit thicker around the edges to form a rim.) Place on a piece of lightly oiled parchment paper for easy transfer to the grill. Repeat with second round of dough.
Brush one side of each dough round with olive oil, and place, oil sides down, on well-oiled grill grate. Close lid, and grill 1 to 2 minutes or until bottom sides are browned and crisp. (Check dough after 1 minute.) Lightly brush tops of dough with oil, and turn. Close lid, and grill about 1 to 1 1/2 minutes, checking after about 45 seconds. Remove dough to a cookie sheet or large flat serving platter.
Spread grilled dough pieces evenly with mascarpone cheese. Top with smoked salmon, and sprinkle with capers and fresh chives. (Pizzas can be served immediately or placed on grill for 30 to 45 seconds, with lid closed, to warm toppings.) Serve pizzas with lemon wedges.