Someone can correct me if I'm wrong. There is a bigger picture to look at than just a cdl. With your truck registered at 11k the total combination gross weight may not exceed 11k so with an empty 15k trailer hook up you are most likely over the registered weight limit. To legally haul more than 11k with the rig you will need a combination registration for the amount of weight you wish to gross.
I know the question is about PA, but I'll spill what I know about WA.
To determine if you need a cdl, you look at the stickers on the door jam, the one the manufacturer puts in with the VIN and tire size and GVWR. If that sticker plus the one on your trailer added together equal more than 26001 lbs you need a cdl.
As far as weight hauled, the towing vehicle gets the tonnage for the trailer also when you buy registration.
So my 98 has a sticker in the door jam for 8,800lbs and my goose neck has a sticker on the tongue for 14,000lbs.
Together they add up to 22,800lbs- I do NOT need a cdl.
I DO need to register my pickup for 22,800lbs to haul at the max weight, as the trailer does not carry tonnage on the registration, it all goes on my pickup.
22,800 is expensive, so I register my pickup for 12,500 (empty weight with trailer) and buy a trip permit for the extra weight when I need it. We still have the trip permits that I can carry around, then fill out the dates when I need it.