1minute 10 seconds.
You toast two pieces of bread at once the first time (30 seconds) When they are both toasted on one side, take one of the pieces off (5 seconds) and put the untoasted piece of bread on the grill (5 seconds). Wait 30 seconds for the last piece to toast.
The riddle doesn’t say that you have to toast both sides of the bread. Instead of taking out both pieces the second time, you can unplug your grill (assuming the grill is electric).
Arguably, the shortest amount of time is 1 minute 13 seconds. You would use the added 3 seconds to flip the piece of bread that stays on the grill and make sure that it doesn’t burn.