In a large shallow dish, beat eggs with a fork. Beat in liqueur, sugar, large pinch of salt, and orange zest and juice. Add milk and beat until combined.
In a large heavy skillet, heat 4 tablespoons oil and 2 tablespoons butter over medium-high until sizzling. Gently dip brioche, one slice at a time, into egg mixture to coat and add to skillet (do not crowd pan). In batches, cook until golden brown on both sides, 5 minutes per batch, flipping once. Transfer to a paper-towel-lined wire rack. Repeat with remaining brioche, adding remaining oil and butter as needed. (Reduce heat if brioche is browning too quickly.) Serve with butter and sugar or maple syrup.
Martha's Tips: Always zest your citrus before juicing -- it's much easier. Dip the bread just before cooking, and take care with tender brioche slices. If the pan overbrowns, wipe it out with a damp paper towel and start again with new oil and butter.