We demonstrate a new connection between e-graphs and Boolean circuits. This allows us to adapt existing literature on circuits to easily arrive at an algorithm for optimal e-graph extraction, parameterized by treewidth, which runs in $2^{O(w^2)}\text{poly}(w, n)$ time, where $w$ is the treewidth of the e-graph. Additionally, we show how the circuit view of e-graphs allows us to apply powerful simplification techniques, and we analyze a dataset of e-graphs to show that these techniques can reduce e-graph size and treewidth by 40-80% in many cases. While the core parameterized algorithm may be adapted to work directly on e-graphs, the primary value of the circuit view is in allowing the transfer of ideas from the well-established field of circuits to e-graphs.