  Recent gravitational wave detections from black hole mergers have underscored
the critical role black hole perturbation theory and the Teukolsky equation
play in understanding the behaviour of black holes. The separable nature of the
Teukolsky equation has long been leveraged to study the vacuum linear Teukolsky
equation; however, as theory and measurements advance, solving the sourced
Teukolsky equation is becoming a frontier of research. In particular,
second-order calculations, such as in quasi-normal mode and self-force
problems, have extended sources. This paper presents a novel method for
analytically separating the Teukolsky equation's source, aimed to improve
efficiency. Separating the source is a non-trivial problem due to the angular
and radial mixing of generic quantities in Kerr spacetime. We provide a
proof-of-concept demonstration of our method and show that it is accurate,
separating the Teukolsky source produced by the stress-energy tensor of an
ideal gas cloud surrounding a Kerr black hole. The detailed application of our
method is provided in an accompanying \textit{Mathematica} notebook. Our
approach opens up a new avenue for accurate black hole perturbation theory
calculations with sources in both the time and frequency domain.
