Last night was my third try at Chicago-style deep dish in my Woodstone, and unless I just got lucky last night I've finally figured out how to do it in there. I didn't do much differently than the book calls for, but here's what I changed:
First, instead of adding the fats to the mixer, I cut my butter and lard into small pieces, and chilled them for 30 minutes. Then I used a pastry cutter to incorporate the butter and lard into the flour.
Secondly, I rolled the dough out much thinner than I had rolled it previously.
Finally, I bumped the Woodstone temp up to 525, turned the radiant flame WAY down (I'm using gas and not wood), used an infrared thermometer to find a spot on the oven floor that was 500 degrees, and cooked the pies on that spot for 28 minutes, rotating 1/4 turn every 7 minutes.
The pies came out nicely browned on the bottom, and the crust was very light and just slightly crispy. From now on this is how I'll do my deep dish pies. Thanks to everyone on here for their assistance. Especially to Raj and Tony!