People in Madrid often come to Plaza Mayor for the calamari sandwiches. The most traditional bars are restaurant is Cervecería Plaza Mayor (in the very same Plaza Mayor), La Campana (Botoneras st 6, going out from the square) or Bar Postas (Postas st 13, another small street going out from plaza Mayor). If you're feeling brave and want something more authentic albeit very greasy, go to Casa Rua (Calle Ciudad Rodrigo 3) for one of their famous "bocatas de calamares" (squid sandwich) for 2.20€.