In a large nonreactive pot, combine 2 quarts cold water with the olive oil, garlic, thyme, salt, and lemon slices. Bring to a boil over high heat and then lower to a gentle simmer.
To prepare the artichokes, peel away the darker outer leaves until you see the pale green-yellow of the tender inner leaves. If using small artichokes, trim off about 1/2 inch of the tops and trim the tough end from the stem. With a sharp-edged spoon, scrape out the choke, if there is any. If using large artichokes, slice off the upper two-thirds of the artichoke and cut off the stem. Pare away the remaining tough leaves from the base. Cut in half lengthwise and scrape out the choke fibers.
Prepare a charcoal or wood fire. Immerse the artichokes in the pot with the simmering brine. Simmer gently until cooked through and fork-tender, 15 to 20 minutes. Drain the artichokes on paper towels and set aside until cool enough to handle.
When the grill is hot, for small artichokes, open out the leaves and press them onto the grill so that they flatten slightly. Cut larger artichokes into quarters and then grill. When nicely browned, remove them from the grill and season with lemon juice and salt.