Ultegra on Trek Crockett 5
Answering this: 'but a full Ultegra system 2x11 can be placed on this frame, right?'

Yes. Frames are not made specifically for either SRAM or Shimano groupsets.

As mentioned in comments you will have to swap out the shifters, rear derailleur and hydraulic brake calipers. Shimano rotors will be compatible I believe.

For the 1x option the SRAM crank can be left in place. The only difficulty I see is that I'm not sure if Shimano makes an Ultegra non-shifter front brake lever unit. You could use a regular shifter/brake unit of course. 

For a 2x11 you will need a clamp on front derailleur, and a compatible crankset. It does not have to be a Shimano unit, you could use another vendors products such as FSA.

One thing to consider. Swapping out a whole groupset is expensive (taking into account cost of components and installation). You may find that selling the Crocket and buying a Ultegra equipped bike is the better option.