Quasi-empirical method usually refers to a means of choosing problems to focus on (or ignore), selecting prior work on which to build an argument or proof, notations for informal claims, peer review and acceptance, and incentives to discover, ignore, or correct errors.
Since it is not possible to find all counter-examples to a theory, it is also possible to argue that no science is strictly empirical, but this is not the usual meaning of "quasi-empirical".
Albert Einstein's discovery of the general relativity theory relied upon thought experiments and mathematics.
Thought experiments are almost standard procedure in philosophy, where a conjecture is tested out in the imagination for possible effects on experience; when these are thought to be implausible, unlikely to occur, or not actually occurring, then the conjecture may be either rejected or amended.
Logical positivism was a perhaps extreme version of this practice, though this claim is open to debate.