Markov chain decomposition is a tool for analysing the convergence rate of a complicated Markov chain by studying its behaviour on smaller, more manageable pieces of the state space. Roughly speaking, if a Markov chain converges quickly to equilibrium when restricted to subsets of the state space, and if there is sufficient ergodic flow between the pieces, then the original Markov chain also must converge rapidly to equilibrium. We present a new version of the decomposition theorem where the pieces partition the state space, rather than forming a cover where pieces overlap, as was previously required. This new formulation is more natural and better suited to many applications. We apply this disjoint decomposition method to demonstrate the efficiency of simple Markov chains designed to uniformly sample circuits of a given length on certain Cayley graphs. The proofs further indicate that a Markov chain for sampling adsorbing staircase walks, a problem arising in statistical physics, is also rapidly mixing.