Transcription factors (TFs) bind regulatory DNA to control gene expression, and mutations to either TFs or DNA can alter binding affinities to rewire regulatory networks and drive phenotypic variation. While studies have profiled energetic effects of DNA mutations extensively, we lack similar information for TF variants. Here, we present STAMMP (Simultaneous Transcription Factor Affinity Measurements via Microfluidic Protein Arrays), a highthroughput microfluidic platform enabling quantitative characterization of hundreds of TF variants simultaneously. Measured affinities for ~210 mutants of a model yeast TF (Pho4) interacting with 9 oligonucleotides (>1,800 Kds) reveal that many combinations of mutations to poorly conserved TF residues and nucleotides flanking the core binding site alter but preserve physiological binding, providing a mechanism for mutations in cis and trans to rewire networks without insurmountable evolutionary penalties. Moreover, biochemical double-mutant cycles across the TF-DNA interface reveal molecular mechanisms driving recognition, linking sequence to function.