Rinse the potatoes and put them in a large saucepan. Cover them with cold water, add 1 tablespoon of salt, and bring to a boil. Simmer uncovered for 15 to 20 minutes, until they are just tender. Drain the potatoes in a colander and place a kitchen towel on top, allowing them to steam for 5 to 10 minutes.
For the aioli, tear the slice of bread into pieces and place in a bowl. Pour the vinegar over the bread, and set aside for 5 minutes.
Place the garlic cloves, egg yolks, zest, juice, saffron, 1 1/2 teaspoons salt and 1/2 teaspoon pepper in the bowl of a food processor fitted with a steel blade. Add the bread, and puree into a paste. With the processor running, slowly pour the oil into the bowl through the feed tube and process until it is the consistency of thick sour cream. Place in a serving bowl.
Slice the potatoes in half and place them on a serving plate. Sprinkle with salt and pepper, and decorate with the chopped chives. Serve with the aioli.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.