How can I gracefully create an opportunity to develop a relationship?

I have a doctor I really admire. She has been tremendously good and generous not to mention helpful tome. She literally saved my life or at least bought me months more of quality of life.

I am a little in awe of her. She is a great role model. But my nervousness seemingly created separation. I feel like I missed an opportunity to know her personally. She gave me hugs called me on weekends said she loved me etc. She is older and doesn’t have a family completely committed to her work but sometimes I think it makes her lonely. I wonder now if it is too late or how to go about breaking through some her professionalism since I responded awkwardly and stiffly feeling overwhelmed. Or if it is inappropriate to try. At the time I was dying quite quickly. Maybe it was just sympathy that intensified emotions.

Anyway I want to at least invite her to dinner but I am not sure how. At this point rejection isn’t the concern I would rather have tried and failed then just giving up. I just am struggling to find or create the right moment to ask.

Any thoughts or ideas?