Prepare grill. Place dough on work surface, knead in 1 Tbsp. olive oil and shape into 2 balls. Cover with a towel and let rest for 10 minutes.
Slice onion halves into thin wedge fans. Brush top and bottom lightly with oil. Grill onion, turning once, until charred and just tender, about 3 minutes. Transfer to a plate and reserve.
Roll out and stretch 1 dough ball to 1/4 inch thick. Place on a greased baking sheet and brush top of dough lightly with oil. Place on grill oil-side down. Brush dough with oil, cover and cook until bottom is crisp, about 4 minutes; flip, cover and cook until crisp on both sides and cooked through, about 4 minutes.
Transfer pizza to a board. Stir together crème fraîche and salt and spread half on grilled dough. Repeat with remaining dough and crème fraîche mixture. Top with onion, prosciutto and basil. Cut each pizza into 4 slices and serve.