How to identify the genes that distal enhancers pair?
Check this technique called Chromosome Conformation Capture (3C). Its variants exist such as 4C, Hi-C etc.

Basically, the principle of this technique is based on the physical interaction between the enhancer and promoter that is bridged by a transcriptional modulator(protein). 


The chromatin is crosslinked
DNA is sheared
The protein is pulled down (Optional)
The fragments of the promoter and enhancer are ligated together and then sequencing/qRTPCR is done. 


Sequencing (low throughput- sanger) can be used to identify enhancers of a given promoter. qRTPCR can be used to quantify the strength of the association.

What I just said is not a high throughput technique. Hi-C is high throughput variant of 3C. See the protocol here.