In generic many-body quantum systems, and at finite energy densities, it is not possible to efficiently resolve individual eigenstates. Even with a fault-tolerant quantum computer, the time required grows exponentially with the number of degrees of freedom. This is because the level density is itself exponential. I will introduce a quantum algorithm that creates quantum states with narrow energy width. In time scaling polynomially with the number of degrees of freedom, this algorithm allows us to achieve inverse polynomial energy width, and with inverse polynomial error. I will then show how our algorithm can be used as the basis for an efficient test of the eigenstate thermalization hypothesis (ETH) in large systems. Interestingly, if our system is described by the ETH, the states we create act as an indirect probe of the eigenstates themselves.