Use a larger chef's knife to cut one of the peach halves into thick slices; lay the peach slices on their sides and cut them, lengthwise, in half again. Cut the peach slices into tidbits, or smaller pieces. Place these small pieces into a smaller bowl and set aside.
Slice the remaining peach halves for the puree.
Place the peach slices into a saucepan, then grate the ginger root into the pan. Add the cinnamon, lemon juice, and brown sugar to the pot and bring this mixture to a simmer over medium-low heat. Cook this mixture until the peaches begin to soften and fall apart- about 10 minutes.
Remove the pot from the stove and puree the peach mixture until smooth using an immersion blender (or transfer to a blender). Allow this puree to cool to room temperature.