Combine the half and half, whole milk and sugar in a medium sauce pan over medium heat. Stir often to dissolve the sugar and prohibit the mixture from burning. Once mixture is heated and sugar is dissolved turn off the heat.

Add the cinnamon sticks to the sauce pan and allow to steep for 1 hour.

After 1 hour, remove and discard the cinnamon sticks. Pour the cooled mixture into a blender along with the lemon juice, mascarpone cheese and very ripe peaches with any juice released from peeling them. Puree in the blender until smooth. Refrigerate the mixture until chilled; 2-3 hours.

Process the ice cream mixture in an ice cream machine following the manufacturer’s instructions.

While the ice cream is processing, grill the peaches. Heat the grill to medium-high (350-375 degrees). Clean and oil the grill. Brush both sides of the peach slices lightly with olive oil and grill for 2 minutes per side. Remove and allow to cool slightly (about 5 minutes). Chop into small pieces to go into the ice cream mixture.

When the ice cream is almost done, add the chopped grilled peaches to the machine and allow to fully mix into the ice cream. Remove the ice cream and spoon it into a freezer-safe container. Allow to freeze fully for several hours or overnight. Additional peach wedges can be grilled the same as the slices to garnish the ice cream when served.