For the lamb: Preheat the oven to 250 degrees F. Arrange an oven rack in the middle of the oven.
In a mortar and pestle or food processor, combine the olive oil, rosemary leaves, garlic, and a pinch of kosher salt.
Pat the chops dry, and place fat-side up on large rimmed baking sheet. Score the fat cap to allow the seasoning to penetrate. Sprinkle the lamb with pepper and salt, being careful as there is salt in the herb paste. Pour the garlic-rosemary mixture over the chops and rub to coat. Let stand for 1 hour or cover and refrigerate overnight.
Place the lamb rack bone-side down (fat-side up) on the pan. Roast until a meat thermometer inserted into the thickest part of the meat reads 125 degrees F, 35 to 40 minutes. The lamb should be slightly undercooked since it's being finished on the grill. Remove from the oven, and set aside to cool.
For the strawberry mint sauce: Meanwhile, place the strawberries in a pot or large saucepan and fill with water to come about halfway up the sides. Bring to a boil over medium-high heat, then stir in the sugar and turn down to a simmer. Simmer until the liquid in the pan is reduced just to coat the back of a metal spoon. We're looking to make a thick strawberry stew.
Transfer to a blender or food processor and blend until smooth-ish. We don't want a puree, but we don't want it to be all chunky monkey. Remove from the heat, add the mint and stir to combine. Pour into a bowl, and set aside until ready to serve.
Preheat the grill to high. Cut the lamb into double chops. When the grill is hot, lightly brush it with some vegetable oil on a rolled towel. Grill the chops, being careful not to overcook; we're just looking for grill marks. We're going for medium rare. I like to call it RARE-DIUM. Plate them family style and cover with foil wrap to keep them warm, and let rest.
Wipe the grill clean with an oiled towel. Grill the lemons cut-side down until the edges are charred and the insides are jammy, about 3 minutes. Serve on a platter with lamb chops, passing the strawberry mint sauce on the side.