Green garlic bulbs look like scallions—they're available at many farmers' markets in the spring. If you can't find them to make this Italian appetizer, substitute the peeled cloves from a head of regular garlic and the green top of a scallion.
Combine oil, garlic bulbs, and rosemary in a small saucepan; bring to a simmer over medium. Reduce heat to low; simmer until garlic is golden, about 15 minutes. Remove from heat; let cool.
Squeeze 2 lemon halves into a large bowl of water; add rinds. Working with 1 artichoke at a time, remove tough outer leaves closest to stem end, then trim pointy tips of remaining outer leaves with kitchen shears. With a serrated knife, cut off top 1 inch; with a paring knife, trim and peel stem. (Transfer to lemon water as you work to prevent discoloration.)
Set a steamer basket in a large pot; add enough water to reach just below basket and bring to a boil. Squeeze remaining 2 lemon halves into pot. Add artichokes, season with salt, and cover. Steam until hearts are easily pierced with the tip of a knife, 30 to 35 minutes. Transfer to a plate; let stand until cool enough to handle, then cut in half lengthwise and scoop out and discard fuzzy chokes.
Strain infused oil; discard rosemary. Mash garlic bulbs into a paste and stir into goat cheese with 1/4 cup infused oil, green garlic tops, and red-pepper flakes. Season with salt.
Heat grill to high. Brush artichokes with infused oil; season with salt. Grill, brushing with more oil and turning a few times, until charred in places, 8 to 10 minutes. Grill bread, turning once, until charred in places, 1 to 2 minutes. Lightly brush bread with infused oil.
Remove artichoke leaves from bottoms; cut bottoms into slices and reserve leaves for eating separately. Slather grilled bread with goat-cheese mixture and top with sliced artichokes. Drizzle with more infused oil, sprinkle with more red-pepper flakes and green garlic tops, and serve.