Preheat the oven to 350F/175C. Place the graham crackers into a food processor and process until they've completely broken down into fine crumbs.
Put the butter into a microwave-safe bowl and microwave for 60 seconds or until melted. Stir the melted butter into the cookie crumbs until it's been totally absorbed.
Scoop the buttery cookie crumbs into a 12cm/5" round greased cake pan and press down firmly with the back of a dessert spoon to form the base. Set to one side.
Quickly rinse out the inside of the food processor and add in the cream cheese, yogurt, sugar and vanilla extract. Process for 60 seconds until combined.
Crack in the egg and pulse 3-4 times until it's just incorporated. Don't over-mix or the cheesecake will crack in the oven.
Scoop the cheesecake filling on top of the base and bake in the pre-heated oven for 30 minutes. After 30 minutes, turn the oven off but leave the door closed and leave the cheesecake inside for a further 30 minutes.
After a total of 60 minutes, remove the cheesecake from the oven and leave on the counter for at least 4 hours to cool to room temperature. Cover with plastic wrap and chill in the fridge for at least 8 hours before cutting into 6 slices.
The cheesecake will keep perfectly well in the fridge for up to one week or in the freezer for up to six months.