As part social scientist by training, I'm partial to the questionnaire idea. If your organization is large enough and management is willing to support your efforts, it may be possible to do a double blind experiment to see if a particular dojo helped the participants in that particular subject matter (As Ewan mentioned, we currently lack an objective method to determine how good someone is at programming overall).
- Determine the topic of your first dojo.
- Create a brief survey/quiz on the key concepts. Ideally the questions will be verifiable enough that the answers can be easily determined to be correct or incorrect, but difficult enough that most participants will not get all the answers correct without the dojo.
- Get as large of a group of volunteers/conscripts as possible to give yourself a lower margin of error.
- Randomly assign half to the control group, which will not initially attend the dojo. Give this group the survey right away. Allow anonymous results to be submitted, all you need to keep track of is if the participant is in the control or not (Two different Google Forms etc. with the same content might be an easy way to set this up).
- Give the first dojo lesson to the other half. Either at the end of the session or the next day, give them the survey.
This will give you the data to do a statistical analysis on the two groups. If the group that attended the dojo saw a statistically significant improvement over the control, we can conclude that the developers at minimum became more competent in the subject area in the short term.
As a bonus, if you would like to see if this information was retained longer term, create another quiz about the same topics, yet is different enough that remembering the answers to the first will not directly assist them. A few months later, give the new quiz to both groups. You can then compare the dojo group with the control to see if the results are still significant.
This would certainly be overkill to do for every dojo, but if it turns out that you can prove that one dojo has a positive impact on one area of programming, we can assume that others will as well.