Using paper towels, dry the birds on all sides as much as you can. Doing so will help you get a strong grip on the chicken when spatchcocking.
Next, spatchcock the cornish hens. Place the hen breast-side down on a cutting board. Find the backbone that runs down the center of the hen. Using poultry shears, remove the backbone starting from the tail and cutting along one side of the backbone. Repeat the same process by cutting on the other side of the bone. Remove it and discard (or save it for making chicken stock later.)
Turn the bird around so that the breast side is up. Locate the breastbone that runs down the middle of the bird. Firmly press down with the palms of your hands until the hen lies flat. Repeat the same process with the second cornish hen. Set them aside.
To make the marinade, whisk together the olive oil, lemon zest, lemon juice, garlic, thyme, paprika, salt, and pepper in a large mixing bowl. I recommend using a bowl large enough that can accommodate both birds. You can use a Ziploc bag to marinade the birds, as well.
Transfer 2 tablespoons of the marinade into a small jar and keep it in the fridge until ready to use. We will use this to brush cornish hens after they are finished cooking on the grill.
Place the birds into the marinade (ensure that they are fully covered with it), cover with plastic wrap, and marinate for at least 30 minutes or for up to 8 hours in the fridge.
When ready to cook, remove the chicken from the fridge while you are preparing the grill. Preheat a gas grill to 350 degrees F.
Clean grill grates. Dip a wad of paper towel into olive oil and then generously wipe the grates.
Remove the cornish hens from the marinade (be sure to shake the excess marinade), and place them skin-side up on the grill. Cover and grill for 10 minutes.
Flip the hens, cover, and grill for 7-10 more minutes or until the skin is crispy and turns golden brown. Keep a close eye on them as they are grilling to avoid burning. Check doneness by inserting an instant read thermometer into the thickest part of the thigh. If it registers 165 degrees F, your cornish hens are cooked.
Transfer onto a plate and cover with aluminum foil. Let them rest for 10 minutes.
Brush the skin of each hen with the reserved marinade. Sprinkle with chopped fresh thyme and serve on a serving platter with lemon wedges on the side.