Mix the mayonnaise, yogurt, garlic, olive oil and salt & pepper in a small bowl.
Spread the yogurt mixture onto each slice of bread.
Layer the half of the prosciutto on top of the yogurt, then half of the persimmon, half of the blue cheese, then the basil. Top with the other slice of bread.
Repeat with the other roll.
Press both sandwiches in a panini maker or in a hot pan until the cheese melts and starts oozing out the sides.