I find "yeux doux" translated in several sources as "goo-goo eyes" or "googly eyes" in English, but none as "doe's eyes".
Popular songs in English refer to "making eyes at" or "having eyes for" as a way of showing love. ("Mom, he's making eyes at me", "I only have eyes for you.") It's a bit old-fashioned, however, not in everyday use any more.
One can also make "sheep's eyes" in English to express love.
And a beautiful woman can be "doe-eyed".
However, I don't find any reference in English sources to "making doe's eyes", and I've never heard the expression myself. Perhaps it is a literal translation of a French expression.
An approximate translation of the sentence might be: "Making eyes at your screen all day will get you nowhere. If it worked we would know it." It's a challanging sentence to translate into coherent English.
Very interesting comments...
I am not sure these explanations are correct.
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