In Natural Language (NL) applications, there is often a mismatch between what the NL interface is capable of interpreting and what a lay user knows how to express. This work describes a novel natural language interface that reduces this mismatch by refining natural language input through successive, automatically generated semi-structured templates. In this paper we describe how our approach, called SKATE, uses a neural semantic parser to parse NL input and suggest semi-structured templates, which are recursively filled to produce fully structured interpretations. We also show how SKATE integrates with a neural rule-generation model to interactively suggest and acquire commonsense knowledge. We provide a preliminary coverage analysis of SKATE for the task of story understanding, and then describe a current business use-case of the technology in a restricted domain: COVID-19 policy design.