Kostant's realization of the Toda lattice has given rise to a fascinating hybrid of ideas from symplectic geometry, algebraic geometry, and representation theory. A modern example is the appearance of this Kostant-Toda lattice in calculations related to the quantum cohomology of the flag variety. It is in this setting that one compactifies the leaves of the Kostant-Toda lattice, thereby constructing a certain class of Hessenberg varieties. It is then reasonable to expect that the Kostant-Toda lattice can be defined on (the total space of) a family of Hessenberg varieties. I will show this to be the case, defining all pertinent objects along the way. I will also emphasize the roles played by Slodowy slices and Mishchenko-Fomenko polynomials. This represents joint work with Hiraku Abe.