Preheat an oven. Pat the Cornish hens dry on all sides with a paper towel, then sprinkl with Kosher salt and freshly ground black pepper.
In a medium bowl, combine unsalted butter, olive oil, the zest of lemon, chopped rosemary, chopped sage, and chopped garlic. Stir until well mixed.
Rub the mixture evenly over the Cornish hens. Stuff the cavity with lemon wedges, shallots, and rosemary or sage. Truss the legs of each Cornish hen. Tuck the wings.
Place the lemon rounds on the bottom of the roasting pan, add the rosemary, sage, and/or thyme sprigs on top. Arrange the seasoned Cornish hens on top.
Transfer to an oven and cook, until the internal temperature into the thickest part of the bird registers between 160-165F. Remove from the oven.
Serve immediately with a sprinkle of flaky sea salt on top.