Peel and cut 1 pineapple into 2-inch cubes. Using a vegetable juicer, juice the cubed pineapple, (you can also use a food processor to puree the pineapple and press it through a strainer lined with cheesecloth so that you are left with pure juice). Pour the juice into a saucepan with the cardamom. Bring to a boil. Remove from the heat, set aside, and let the cardamom infuse.
Peel the remaining pineapple and cut horizontally into 1-inch round slices. Cut out the center core with a small size round cutter. Place slices of pineapple in the pineapple juice, bring to a boil, and then turn off the heat. Let cool and marinate overnight in the refrigerator.
The following day, preheat oven to 350 degrees F. Remove pineapple slices from the juice place on a sheet pan. Roast in the oven until tender, about 30 minutes.
To plate, place warm pineapple in a bowl and top with a scoop of Green Tea Melon Granita.
Green Tea Melon Granita:
Place the melon in a bowl. Add the ground ginger, green tea powder, and honey. Blend with an immersion blender until smooth. Pour into a saucepan and bring to a boil. Remove from heat and strain into a shallow 9 by 13-inch metal pan or dish.
Stir in the lime juice, let cool, and then freeze overnight.
The following day, remove the granita from the freezer, scrape with a fork to make the granita, and return to freezer until ready to serve.