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Squeeze juice from halved lemons into large bowl filled with cold water. Cut stem off 1 artichoke, leaving about 1”. Snap off outer 2 rows of leaves. Cut off top third of artichoke. Quarter artichoke lengthwise. Using small knife, cut out choke and prickly small leaves. Place artichoke in lemon water. Repeat with remaining artichokes.
Bring large pot of salted water to boil. Drain artichokes from lemon water, add to pot, and boil until crisp-tender, about 15 minutes. Drain, and transfer artichokes to rack and cool. (Artichokes can be prepared 1 day ahead. Cover and refrigerate.)
Combine 2 tbsp. olive oil with balsamic vinegar and mustard in a small bowl. Season with salt and pepper, to taste. Set aside until ready to serve.
Heat grill to medium-high heat. Brush artichokes with oil and season them with salt and pepper, to taste. Grill until tender and lightly charred in spots, turning occasionally, about 8 minutes. Transfer artichokes to a serving platter and drizzle with Balsamic Mustard Sauce. Serve warm or at room temperature.
(Optional: If you don't have a grill or weather not permitting. You can roast at 400 degrees until tender and light brown. Interior of artichoke facing up while roasting.)