Dear Ms. Crawford,
Could I be so bold as to ask if your family might at home some night in this coming week, and if there was some evening when we might have a moment to speak? I would understand if this would not be something you would care to do, and if that is the case then please do not be afraid for my friendship, but if you think a conversation of the most serious nature would be amenable to you then I would look forward to speaking with you then.
A conversation of a serious nature has me quite curious as is my family. Having you join us for Saturday dinner would work splendidly for us and would not interfere with your working hours. We also do not have any social obligations on that evening so please let us know if this time and day works for you.