Position rack in center of oven and preheat oven to 425°F.
Arrange a large wire rack over a large rimmed baking sheet or shallow roasting pan. In a small bowl, combine the pesto and goat cheese, mashing it with a fork until evenly mixed.
Place a game hen, breast-side down, on a cutting board. Remove backbone by cutting along both sides with poultry shears. Flip the hen over and flatten it by pressing down on the breast bone with your palms. Using a chef's knife, split the hen in two along the breastbone. Extend the wings on each side, and chop off the last two joints. Remove and discard any large deposits of fat. Repeat with remaining hens.
With your fingers, loosen the skin over the breast and leg. Insert a heaping tablespoon of the pesto mixture under the skin and work it so it evenly covers the breast and leg meat. Smooth the skin back to its original position and repeat with remaining hens. Can refrigerate hens for 2 hours if desired.
Arrange hens on the large wire rack. Do not overlap. Brush the surface of the hens with olive oil and sprinkle with salt and pepper.
Roast hens, rotating pan about halfway through, until their nicely browned and juices run clear when a thigh is pricked with a skewer or toothpick (about 30 minutes. Cover loosely with foil and allow to rest about 5 minutes before serving.