Brush 1 side of each tortilla with oil.
Place tortillas, oiled sides down, on a baking sheet.
Top half of each tortilla with tomatoes, asparagus, and basil; sprinkle with vegan mozzarella.
Fold tortillas in half over the filling, pressing lightly.
For a charcoal or gas grill, preheat to medium heat.
Grill the quesadillas on the rack of an uncovered grill for about 2 minutes, or until the vegan cheese starts to melt and the tortillas begin to brown, turning once halfway through.
To serve, cut the quesadillas into wedges. If desired, serve with fruit salsa or marinara sauce.