For lemon and thyme marinade, combine all ingredients in a non-reactive bowl and season to taste.
Add quail to marinade, turn to coat evenly, cover with plastic wrap and marinate in refrigerator for 30 minutes. Meanwhile, pat vine leaves dry with absorbent paper.
Working with 2 leaves at a time, overlap leaves on a work surface and place one quail half on top, leaving some of the breast and leg extended past the vine leaves. Fold leaves over to enclose and set aside. Repeat with remaining quail and vine leaves.
Preheat a chargrill pan over high heat. Cook quail in batches, turning once, until just cooked through (2 minutes each side).
Meanwhile, for grape and hazelnut vinaigrette, combine ingredients in a bowl, season to taste and toss to combine. Spoon over quail, crumble over the feta and serve with lemon wedges to the side.
Note Vine leaves in brine are available from select supermarkets and delicatessens.