First, grill the peaches. You can do this on an outdoor grill or use a grilling pan indoors. I advise brushing the grill or pan lightly with the vegetable oil {or other neutral-tasting oil} to be sure that the peaches won't stick!
Heat pan/grill to medium heat, and grill the peach slices about 3-4 minutes on each side, until softened and with evident grill marks. Set aside on a plate, and sprinkle with the cinnamon and sugar. Set aside to cool, then cut into tiny pieces.
In a large bowl, beat the heavy whipping cream to stiff peaks with a hand mixer. When cream is whipped, slowly drizzle in the condensed milk until completely incorporated. Beat in the vanilla and almond extracts.
Grab a 9x5 loaf pan to make your ice cream in! Pour in about ⅓ of the whipped cream mixture, then sprinkle on ⅓ of the peach pieces and ⅓ of the amaretto cookies, crushing them lightly in your hands first. Repeat for two more layers.
Wrap the pan tightly with plastic wrap, and freeze until solid, at least 6 hours. Enjoy!