Does personality imply an inclination to be an expert in one field or average in "all" fields? The question of whether it is better to be an expert in one domain or average in many is too vague.

However, instead, I'd like to focus on the following:


Are individuals born to prefer being an expert in one domain or average in many domains? 
Is there any serious research on this? 


For example I'd very like to be an expert in swimming but after some time of hard training this activity becomes boring for me. I need to run, I need to play ball team games etc. Conversely, Michael Phelps, world swimmer, incline to be an expert in one branch.   

We can talk about branch of study, hobbies, sport, social interactions, languages, etc.
I imagine there are many ways of looking at this question. Here are just a few ideas:

Society and specialisation:
One lens for viewing this question is to focus on the reward structure of our society. There are many forces in society which encourage specialisation and the development of specific expertise. Careers are typically built around developing expertise in a particular domain (e.g., doctors, lawyers, computer programmers, etc.). The nature of the reward structure that is facilitated by large economic markets means that the most popular actors, musicians, and politicians, and the most successful athletes get more of the financial and social rewards than those that just dabble in such activities. The flipside of a specialised society is that you don't need to know how to do many things, because you just pay someone else to do it. Thus, from this perspective, in historical terms society appears to be encouraging expertise more than in the past.

General individual differences: The concept of need for achievement refers to an "individual's desire for significant accomplishment, mastering of skills, control, or high standards." 

Self-actualising: You could look at research on how people find meaning in their life and how mastery and expertise fits into that experience. In particular, the journey of acquiring mastery in a domain has often been linked to feelings of meaning and purpose. For example, have a look at something like Self Determination Theory and the role of competence in achieving a sense of personal growth.

Expertise and deliberate practice research: There's a substantial body of research that has studied people acquiring expertise in particular domains. Ericsson et al (1993) provides a good introduction to this research. A central claim in Ericsson's model is that deliberate practice is key to acquiring expertise. And given that deliberate practice is optimised for improving performance it may not necessarily be intrinsically enjoyable. Thus, this suggests that the motivation to engage in deliberate practice over an extended period of time may be a key differentiator between those who do and do not acquire expertise. 

References

Ericsson, K. A., Krampe, R. T., &amp; Tesch-Römer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological review, 100(3), 363. PDF