This paper presents a new approach to two-level hazard-free logic minimization in the context of extended burst-mode finite state machine synthesis targeting generalized C-elements (gC). No currently available minimizers for literal-exact two-level hazard-free logic minimization of extended burst-mode gC controllers can handle large circuits without synthesis times ranging up over thousands of seconds. Even existing heuristic approaches take too much time when iterative exploration over a large design space is required and do not yield minimum results. The logic minimization approach presented in this paper is based on state graph exploration in conjunction with single-cube cover algorithms, an approach that has not been considered for minimization of extended burst-mode finite state machines previously. Our algorithm achieves very fast logic minimization by introducing compacted state graphs and cover tables and an efficient single-cube cover algorithm for single-output minimization. Our exact logic minimizer finds minimal number of literal solutions to all currently available benchmarks, in less than one second on a 333 MHz microprocessor-more than three orders of magnitude faster than existing literal exact methods, and over an order of magnitude faster than existing heuristic methods for the largest benchmarks. This includes a benchmark that has never been possible to solve exactly in number of literals before.