In a medium saucepan, cover potatoes and garlic with water and season generously with salt. Bring to a boil, then reduce heat and simmer until potatoes are tender, 8 to 10 minutes. Drain, reserving 1 cup cooking water, and return to pan.
Add sardines and their oil. Whisk mixture until combined, then gradually whisk in oil. Add reserved cooking liquid to adjust consistency. Stir in lemon zest and juice and season to taste with cayenne.
Preheat broiler. Brush bread with oil and toast until golden brown and crispy, 1 to 2 minutes. Serve warm with brandade.