Pinti
Pinti is not strictly Leccese, its actually a Neapolitan Pasticceria, and has a dangerous selection of Neapolitan cakes and sweets which, if you wanna max your daily calorie limit in about 15 minutes, will blow your mind.
The interior is a simple bar/counter (cassa left / sweets & cakes middle / coffee bar right), tastefully littered with neo-antique Italian furniture (Pinti is not actually old, I think it opened in 2017). The team are generally helpful and speak plenty of languages between them.
Outside, Pinti comes into its own as it spills out onto the large paved space in front of Porta San Biagio. Here I have whiled away many many MANY mornings with a coffee & pasticciotto, reading my book or relaxing in the sun.