Heat a cast-iron grill pan over medium. Place shrimp on grill pan; cook, turning once, until shrimp are cooked through and no longer opaque, 2 to 3 minutes per side. Transfer to a cutting board. Let cool until easy to handle, about 5 minutes.
Separate shallot into rings, and add to a medium bowl. Add lime juice, and toss. Let stand until slightly softened and mellowed in flavor, about 2 minutes; drain.
Peel shrimp, and discard shells. Cut shrimp into 1- to 1 ½-inch pieces. Add shrimp and lychees to bowl with shallot. Add nam yum, and toss to combine. Chill, uncovered, until shrimp absorbs flavors, about 5 minutes.
