Must eat resturants in Paris and Rome?

Paris: Ciel de Paris (55th floor of the Montparnasse, right under the visitor platform)

Le Soufflé

My suggestion does not contemplate food quality only, but also the place...

58 Tour Eiffel is a restaurant in the 1st level of Eiffel Tower.
The food is good enough (not superlative, but certainly not bad) but the view is great AND with the reservation, you will skip the queue to enter the tower.

Here´s the link: http://www.restaurants-toureiffel.com/

My current favorite place in Paris is L'Ami Jean -- Basque food. Great fun service, busy, fast, and the cook is from a book (well, his book anyway).  Been there multiple times -- each time the food was amazing.  Must have the rice pudding for dessert.  It's mind boggling good. 

In Rome -- Al Pompiere : Penne Arriabiata 

In the old jewish quarter - Ristorante Piperno for fried artichoke !  

Flavio al Velavevodetto - Cacio e Pepe ... 

Remember, you can always order "une per due" (1 for 2) to split plates so you can order and try more things.  I find 1 pasta dish for 2, then 1 "secundi" (second / meat / seafood) for 2 leaves enough for dessert! :) 

Chez Janou in the Marais

Au Pied de Cochon late at night

My recommendation for non-budget dining in Rome would be Antico Arco and Osteria Barberini. They both have great food and are in very nice locations. If you want more than just food you might want to try social dining too.

Not as sure about Paris, though there's a good overview of the dining scene here: http://wikitravel.org/en/Paris#Eat

I've just come back from Rome. The most amazing dining experience there is at the Rome Cavalieri, Waldorf Astoria Hotels & Resorts. The restaurant is La Pergola, has 3 Michelin stars and the terrace has a panoramic view of Rome. Everything about the experience was excellent and, while it is a splurge, it's worth every cent.