If you want the bottom to be more leopard spotted, you'll need to get the pie off the cardboard and directly onto a pizza stone, baking steel, metal pizza pan or even the bbq grill grates for a few minutes. But getting the uncooked moist/sticky dough off of the cardboard tray is a pain.
Chuck does what PM instructs (below) , which is to start cooking the pie by putting the cardboard on a stone/steel heat shield.
But after a few minutes cook time, the dough will have firmed up enough that it should be easy to slide the partly cooked pie off of the cardboard and onto the stone/steel/pan for browning. Particularly with a kettle that runs very hot, be careful not to ignite the edges of the cardboard. So maybe try a 14 inch pie first with your 16 inch stone.
You'll have to play around with the timing of when to pull the cardboard out. The goal being to get the bottom crispy/spotted (but not burned) by the time the top is cooked.
If you want to the whole cook time to be on the stone (which is what I sometimes do), flip the pie face down (with plastic still on) on a cookie sheet or cutting board. Peel the cardboard off the upside down pie. Brush the underside of the dough with a little olive oil and sprinkle with corn meal. Put your pizza loading device (pizza peel, cutting board, cookie sheet) on top, flip it over, remove plastic from the top. The corn meal should keep the dough from sticking. Then launch pie onto hot stone.
READ FIRST: Protect yourself against charred pizza! Our trays are flammable and must be protected from direct exposure to the heat source. To prevent scorching or igniting the paper tray or Fresh Pan aluminum pan, place it on a heat-shielding surface, such as foil or a pizza stone. Pre-heat pizza stone for 10-15 minutes for best results.