I am an old man of 65, so you may feel free to disregard what I have to say.
I have been interested in romance all my life. And I have gone through the grade school stuff (notes back and forth, and friends bringing messages) to "going steady" in high school, through flirting and dating in college to outrageously romancing as a sailor in the Navy.
Sounds silly to say so, but I've even "enjoyed" silently admiring a long-haired beautiful country singer, looking sad (without her), and inviting her to my table for a drink (and succeeding, by the way).
I quite agree that women do give signals that you may (I stress, may) be able to interpret.
Nowadays, females are more inclined to not wait for the male to initiate the dialogue. They may initiate the signals.
But whether the female does not not, it is my considered opinion that "shyness" or fear of "embarrassment" or fear of "failure" are not nearly as important as the benefits of being with someone you care for and enjoy.
I would recommend - perhaps at the restaurant, telling her that while your conversations and letters with her have been very important to you, and you value your friendship, you realize that you care for her. Ask her if that bothers her.
She may let you down, but I doubt she will laugh at you. She may indicate she only wants friendship.
But, I'd say it will be easier (on your feelings) than trying to measure her pupils.
Or, if you are not a good talker (when together), take her hand, lean across and kiss her lightly. I'd say you'll get some kind of response.
If you go with the last paragraph, take a rosebud with greenery and baby's breath, to give her as you're seated.
Believe me, she'll know you are initiating a romantic dialogue.
