I recommend you taking feedback on several events like "Mango Day". These events cannot be compared with one or two day employee training wherein feedback is taken at the end of the day. This is because these events are of secondary in nature. Therefore, it is better to take collective feedback rather than individual one.
Employee engagement has far more broader scope. Engagement is on three counts. One is physical engagement, second one is mental engagement and third one is about intellectual engagement.
Other than skills required for the job, employees possess certain innate abilities. While they may speak about it with their colleagues at the workplace, they may not get a chance to demonstrate it. Events like "Mango Day" that you organised, help employees to demonstrate these skills. These events give temporary relief to the employees from their routine grind. Employees come together but for different reasons and it helps them to develop a personal contact. Added to this is an element of fun.
Nevertheless, the physical engagement does not serve a purpose beyond a point. Engagement also depends on the workplace environment or culture of the company. Employees vie most for a culture of cooperativeness or culture of collaboration. They also want healthy interpersonal environment. Against this backdrop, a question arises is what if a manager or senior colleague shows overbearing pride, rub others wrong way? Will events like "Mango Day" eradicate disgruntlement originated because of the audacious behaviour of the manager or any other senior person? Do events like these give vent to the suppressed feelings of the employees? Because of these factors, employees may get mentally detached or disengaged.
I have written above paragraph to bring out difference between employee satisfaction and employee engagement. Events like "Mango Day" may satisfy employee but may not engage them per se. As Frederick Herzberg has told us in his theory of motivation, opposite of satisfaction need not be dissatisfaction and real satisfaction or engagement may not get captured because of organisation of these events.
Intellectual engagement is far more higher form of engagement. In the Abraham Maslow's theory of needs, the top most need is self-actualisation need. Intellectual engagement happens when employers fulfil this need of the employee. If this need is fulfilled, then employees do not required any kind of physical or mental engagement. Many start ups have started in car garages or any other cheesy place. Employees worked at such workplaces though their hygienic needs were not fulfilled because fulfilment of intellectual overruled fulfilment of other needs.
Therefore, you may create a questionnaire to measure employees' mental engagement or intellectual engagement with the company.
By the way, in your company you create events to foster personal contact or even attachment. Nevertheless, while seeking query or help, you have preferred to remain anonymous. Does anonymity help engagement with the seniors and that too while seeking help? Can you explain this contradiction?
8th June 2018 From India, Bangalore