Isolating a specific gene (specifically TRAV* series of genes) for sequencing
Restriction enzymes are usually only used when you do molecular cloning using plasmids or other forms of shorter DNA moleclues, since - like you said - using them on genomic DNA would cut it into much more than thousands of pieces.

Instead what you can do is perform your PCR directly on the genomic DNA (or potentially cDNA, if you want to get rid of introns).
Doing PCR on genomic DNA is often a bit tricky (i.e. you'll likely have to try different conditions to get a product), but generally possible. If you want to analyse the whole coding region it might be better to extract mRNA instead of DNA and reverse transcribe it, so that can amplify the coding region as one block, that doesn't contain Introns - PCR on cDNA is also easier than on genomic DNA.

Another thing you need to look out for is that T-Cell-Receptors (similar to antibodies) are different in every T-Cell. This means that you can't use Sanger sequencing, which will only give you a proper result if you input has the exact same sequence. To get the sequences of a mix of T-cells you would need to use next-gen sequencing.