Cook orzo in a pot of salted boiling water until al dente, about 7 minutes. Meanwhile, whisk together 5 tablespoons oil, lemon zest and juice, shallot, and oregano; season with salt and pepper. Drain orzo, transfer to a bowl, and toss with half of dressing.

Season chicken with salt and pepper. Heat a large skillet over medium-high heat. Swirl in remaining 1 tablespoon oil; add chicken in a single layer. Cook, flipping once, until golden and just cooked through, 1 to 2 minutes a side. Transfer to a plate. Add artichoke hearts to skillet; cook, stirring, until warmed through, about 1 minute. Transfer to bowl with orzo. Shred chicken; add to orzo with remaining dressing. Toss to combine; season with salt and pepper. (Store in refrigerator up to 2 days.) Serve in lettuce cups.