Slice 1 in. off leafy top end of artichokes. Trim darkened base of stems with a chef's knife and trim stems' fibrous outer layer with a paring knife. Pull off small, tough leaves and snip off thorny tips with scissors.
In a large pot, bring 1 1/2 in. water to a boil. Add salt, lemon juice and juiced halves, and artichokes. Cover and bring to a boil; then reduce heat and simmer until a knife tip slides easily into artichoke bottoms, 30 to 40 minutes. Drain and let stand until cool enough to handle. Meanwhile, heat a grill to medium (350° to 450°).
Quarter each artichoke lengthwise and scrape out fuzzy centers. Brush all over with olive oil. Grill, covered and turning once, until lightly browned, about 10 minutes total. Serve with sauce.