I would begin with breakfast at Rochelle Canteen on Arnold Circus in Shoreditch (arnoldandhenderson.com). This lovely, understated restaurant with communal tables is housed in a converted bike shed, part of a former primary school now used for studio space.
I would then take a tour of 19 Princelet Street Synagogue (having planned well in advance as it is rarely open due to its fragile condition, 19princeletstreet.org.uk). Once a Protestant Huguenot weaver's house, it has a shabby facade that gives no indication of the beautiful secret Victorian synagogue built by Jewish immigrants over the former gardens of the house.
I would buy a sandwich lunch at the charming A. Gold (agoldshop.com) in Spitalfields: it's a small grocer and wine merchant specialising in traditional British food. Then I would explore the nearby 18th century side streets, with a stop at Hawksmoor's magnificent baroque Christ Church (ccspitalfields.org).
I might shop for vintage clothing on Brick Lane and adjacent streets, or look for current fashion, furnishings and artisanal food at Spitalfields market (spitalfields.co.uk), London's oldest, dating from 1638.
Depending on my mood I would consider dinner at the small modernist Formans restaurant (on the canal in Hackney Wick, overlooking the Olympic site, formans.co.uk) for its renowned smoked salmon. Or, for an entirely different experience, I would go to the over-the-top Tayyabs (tayyabs.co.uk) in Whitechapel for some authentic Indian food.
To end my day, if it were a Monday or Wednesday night, or especially the last Thursday evening of the month, I'd visit Dennis Severs' House (back in Spitalfields, dennissevershouse.co.uk) for the strangest museum experience you are likely to encounter. Severs transformed his Georgian terrace house into the home of a fictitious family of Huguenot silk-weavers. What he called his "still-life drama" is now preserved and run by the Spitalfields Trust. "Silent Night" visits every Monday and Wednesday evening involve a wander through the house's 10 rooms in silence, ending in champagne and a Q&A with curators.
If it were Sunday night, I'd take a Jack the Ripper Walking Tour. It may sound corny, but believe me, it's not.
Perhaps I'd stop off at the Cat and Mutton on Broadway Market (catandmutton.com) for a nightcap before heading home through quieted streets where past and present lie side by side.