One reaction to your suggestion might be that it’s unclear why we’d want to know whether a critical-thinking workshop—whose explicit goal is presumably to improve critical thinking—happens to also enhance “mental health” (a catch-all I’m using for your self-esteem and control over one’s life). I mean, why not just develop separate interventions (e.g., workshops, psychotherapy, self-help materials) to improve these two distinct things without worrying about whether improving one incidentally improves the other?
On the other hand, I can imagine the potential value of showing that improving critical thinking has indirect benefits; as you suggest, this might be useful in convincing a funding agency that such training has a favorable cost-benefit ratio. If you want to actually conduct such an experiment or at least think about some good experimental designs, I’d be interested in discussing plans for such a study. Unless you (or others involved in the discussion) are familiar with the relevant research domains, this would probably entail a substantial amount of reading and searching for relevant studies, which I don’t have time to do myself.
For instance, a good first step would be to check for existing studies similar to what you’re considering, such as in psychology, education, philosophy, and related disciplines. There’s certainly empirical research on aspects of critical thinking, and undoubtedly some of this addresses the impact of training on conventional learning outcomes (e.g., performance on critical thinking tasks). I have no idea, however, whether there’s been work on mental-health outcomes (e.g., subjective well-being, quality of life, stress, autonomy). Similarly, it be worthwhile to look into what types of variables have been shown to relate to your target mental-health outcomes, especially variables that might be manipulated experimentally or that might relate somehow to critical thinking.
It’d also be good to think carefully—and do relevant reading where feasible—about what exactly you mean by “critical thinking,” “workshops,” and “mental health”; each of these key constructs could be operationally defined in various ways, and different operational definitions may yield different experimental results. That is, there are many ways to run a workshop and many ways to specify and measure both critical thinking and mental health; it’s likely that a particular style of workshop aimed at improving a particular kind of critical thinking improves one type of mental health more than another type. In a similar vein, it’d be wise to consider potential mechanisms by which improved critical thinking might also improve mental health. This could help make decisions about the appropriate operational definitions as well as other other decisions in the planning of a good study, such as how long after the workshop to assess outcomes of interest, in what circumstances to assess these outcomes, and whether to manipulate or measure other variables to understand the process (e.g., mediators or moderators).
That’s all preliminary work. Depending on what you learn about these initial considerations, certain types of experimental designs may be more appropriate than others. I won’t go into detail here for now, but some decisions to make involve the number of workshop styles, the number of kinds of critical thinking, whether these two factors will be fully crossed (vs. nesting, fractional factorials, etc.), how many different instructors will lead workshops, the nature of any control conditions (e.g., waiting list, instruction in another topic), the number of measures of mental health (e.g., as manifest indicators of latent variables), what populations will contribute samples, how many participants will be in each condition, whether participants will experience more than one condition (e.g., blocking), the timing of mental-health measurements (e.g., pre-test, post-test, follow-ups), administration of manipulation checks (e.g., measures of critical thinking ability), strategies to avoid respondent or instructor expectancy effects (e.g., blinding, cover story), and anticipated data-analysis techniques. Among other things.