Why Cheese Turns Rubbery in Skillet Cacio E Pepe

I made the Milk Street recipie last night where you do it in a skillet, add less water, etc. I know this is a hard recipe to make and in the end, it did turn out ravishingly delicious. However, there is this one step I am struggling with, which would have made it better, creamier. So, in the first part of the recipe: "To the same skillet, add the spaghetti, placing the noodles parallel to each other in the center of the pan, then add 2 tablespoons of the cheese mixture", and then later, when you add the rest of the cheese to the sauce, both times, a good portion of the cheese globbed up to a rubbery texture. The first time I had to remove it all, the second addition, I was able to tease it back into the sauce for the most part. What am I doing wrong? It wasn't on super high heat. Maybe just a tad too high? Any suggestions?

(Oh - the cheeses I used were some grated parmesan and some parm/asagio/romano "salad blend" that was shaved and I cut it up with a knife to a pretty small size before I added it. Both cheeses were taken from the fridge and brought to room temp before adding.)

Thank you so much for any help with this because I definitely want to do it again.