1. In a 5- to 6-quart pan, combine lemon juice and 2 quarts water. Cover and bring to a boil over high heat.
2. Meanwhile, cut about 1 inch off artichoke tops; discard tops. With scissors, cut off remaining thorny tips on outer leaves and discard. Slice artichokes in half lengthwise through stems. Rinse and drain.
3. Immerse artichokes in boiling water. Return to a boil, cover, and simmer over low heat until artichokes are tender when pierced, about 25 minutes. Drain. Scoop out and discard fuzzy centers.
4. As artichokes cook, rinse fennel. Trim and discard stems and any bruised spots. Break segments apart.
5. In a 2- to 3-cup pan over low heat, melt butter with oil, garlic, and anchovies to make bagna cauda sauce.
6. Put fennel and artichokes on a platter, the bread in a basket, and bagna cauda sauce on an electric or candle warmer; take care not to scorch sauce. Dip fennel and artichoke leaves and bottoms in sauce and hold pieces of bread under vegetables to catch drips as you eat. Also dip the bread in the sauce to eat.