Although I love Anteprima - it may be the place not near home that I dine at the most frequently - I would not consider it "special and romantic". It's loud (the outdoor patio in the rear is better but it's getting too cold for that option) and the tables are thisclose together. Again, I love the place for its consistently outstanding food (and bargain prices), but I'm not sure it works for the stated desires.
I like Lula and think it's a better option.
I can think of other Italian places with great food and a more romantic atmosphere - notably Gioco in the South Loop and Campagnola in Evanston - but I don't know whether either has carrot cake. (Campagnola rotates their desserts and often has the best bread pudding on the planet, if that's of interest.)
Wherever you go, when making your reservation, make sure you mention that it's her birthday.