When I arrive at Palazzo Guicciardini in Florence, tourists are pressed against the iron gates, jostling for a view of this medieval marvel. They aim their phone cameras toward the palazzo’s serene garden, with its meticulous topiary and blooming rose bushes bordered by the Medici’s famous Vasari Corridor.

For Barbara Casasola, a fashion designer who has cultivated a passionate following for her essentialist clothing line, this is not a landmark—it’s home. I squeeze past the crowd of tourists and head upstairs, where she greets me at the door and leads me into her living room. It neighbors the apartment of the Guicciardini descendants; the family has owned this palazzo for hundreds of years.

In the 13th century, these quarters housed the famously humble and beatific St. Philip...