Quasi-empirical methods are applied in science and in mathematics. The term 'empirical methods' refers to experiment, disclosure of apparatus for reproduction of experiments, and other ways in which science is validated by scientists. These are studied extensively in the philosophy of science but are not directly applicable to fields not invalidated by real experiment (mathematics, theology, ideology). In these fields, the prefix 'quasi' came to denote methods that are 'almost' or 'socially approximate' an ideal of truly empirical methods.
It is unnecessary to find all counterexamples to a theory; all that is required to disprove a theory logically, is one counterexample. The converse does not prove a theory; Bayesian inference simply makes a theory more likely, by weight of evidence.
One can argue that no science is capable of finding all counter-examples to a theory, therefore, no science is strictly empirical, it's all quasi-empirical. But usually, the term 'quasi-empirical' refers to the 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. These are common to both science and mathematics - and do not include experimental method.
Einstein's discovery of the General Theory of relativity relied upon thought-experiments and mathematics, and empirical methods only became relevant when confirmation was looked for. Some empirical confirmation was found only some time after the general acceptance of the theory.
Thought experiments are almost standard procedure in Philosophy, where a conjecture is tested out in the imagination for its imagined effects on experience; where these are thought to be implausible, or unlikely to occur, or not actually occurring, then the conjecture is either rejected or amended. Logical Positivism was a perhaps extreme version of this.