I think that GATK alone will not solve the problem of running 3000 samples. You can run scatter-gather on each sample, which could reduce the time for each sample. Ideally, you would want to run this computation on some sort of cluster/super-computer space or on AWS. Then you can run as many computations at the same time as you want. For example, you could run 3000 nodes and then run GATK on each sample on each node and it will take only the time for the longest sample to finish.

http://arvados.org is solving this by creating a completely open-source platform for managing and running computational pipelines (our code can be found here! Through a few clicks, you're able to run your 3000 samples seamlessly and be able to track provenance for each sample.

Have a look at using HaplotypeCaller in GVCF mode (single sample) followed by GenotypeGVCFs. This is a relatively new workflow designed for just such situations as you describe, I think.