eat with eyes is a popular expression that means to eat more than what is needed to go hungry, eat over the limit, overeat. It means that the person is linked to gluttony, that is, he is a glutton, one who eats a lot.
The expression is also used when the individual is simply watching the food closely and for various reasons cannot eat. It is even used when someone is enjoying and desiring an object or an admired person.
Eating with the eyes, costing the eyes of the face, lynx eyes are popular expressions that use the word eyes in a figurative sense.
expression source
The expression originated in ancient Rome, where funeral rites were held in great offerings to the gods. During the ritual it was not allowed to eat, only to admire the food. You ate with your eyes.
Eating with your eyes - blog
The blog eat with your eyes talks about eating, in a relaxed way, with guidelines and clarifications.
To eat with your eyes - book
To eat with your eyes is a book written by Renata Sant'anna, which features photos of various works by various artists who use food, among other materials, in artistic creation.