Smash garlic on a cutting board with the flat side of a chef’s knife, and finely chop.
Finely chop anchovy fillets and mash into garlic until a paste forms.
Pour both oils onto a large shallow platter or rimmed plate.
Stir garlic-anchovy paste, lemon juice, salt, and pepper into the oil.
Stir together chives, parsley, lemon zest, and Parmigiano-Reggiano in a small bowl, and sprinkle evenly over oil mixture.
Serve immediately with warm bread for dipping and scooping.
