Preheat a grill to medium-high. Drizzle 1 tablespoon olive oil on a large baking sheet. Place the 4 dough pieces on the baking sheet and press each out into a 4-by-8-inch rectangle. In a small bowl, stir together the yogurt, scallions and lemon juice.
Pile the corn, jalapeno and cheddar in the center of each piece of dough and press the toppings in. Fold over the dough and press all over to seal, enclosing the filling. Let rest in a warm spot until slightly risen, about 10 minutes.
Meanwhile, arrange the mushrooms upside down and drizzle with olive oil. Place the mushrooms and the sausages on the grill; cover and grill, turning once, until cooked through, about 4 minutes for the mushrooms and 6 minutes for the sausages. Transfer to a board. Thickly slice the mushrooms and slice the sausage on an angle; keep warm.
Press each flatbread again into an 4-by-8-inch rectangle. Grill, covered, until crisp and golden, about 4 minutes on each side.
Spread yogurt sauce on each flatbread and layer with the sausage, mushrooms and tomatoes. Fold the flatbreads over the filling and serve warm.