In a large bowl, mix together the peaches, sugar, lemon juice, and extracts. Let sit for about 15 minutes, stirring occasionally until the sugar is dissolved and peaches have released their juices. Mash with a potato masher. Measure out 2 cups of the mashed peaches. Reserve the remaining mashed peaches in a small bowl and chill until very cold. Puree the 2 cups in a blender or food processor. Add the heavy cream and pour into a clean bowl, cover with plastic wrap and chill until very cold. (If you're in a hurry, you can place it in the freezer for about 20 minutes. Whisk it well in case it freezes around the edge of the bowl.)