Put one cup of the raspberries in a small bowl, sprinkle with two Tbs. of the sugar, mash with a fork, and let sit a few minutes.
Meanwhile, put the remaining 2 cups of berries and 2 more Tbs. of sugar in a food processor (or blender) and process until the berries form a purée. Strain through a fine-mesh strainer into a small bowl, pressing with a rubber spatula to get the seeds out. Squeeze in a few drops of lemon juice and a tiny pinch of salt. Taste and add more sugar or lemon if needed. The sauce should be thin enough to drizzle. If it seems too thick, add a few drops of water. Cover and refrigerate.
In a medium bowl, combine the cream, the crème fraîche, and the remaining 1 Tbs. sugar and whip with a hand mixer until the mixture forms firm, thick peaks. Stir the mashed berries and sugar and lightly fold into the cream mixture with a rubber spatula, leaving streaks.
Reserve 6 of the cookies for decoration and crunch up the rest into uneven pieces—not too small. Fold the cookies into the cream. Cover with plastic wrap, pressing the wrap onto the surface of the cream, and chill until the cookie pieces are thoroughly softened, at least 2 hours and preferably overnight.
To serve, use an ice cream scoop or large spoon to scoop out a mound of cookies and cream into a small bowl or onto a plate. Drizzle a ribbon of raspberry sauce around the plate, tuck a cookie into the cream, and decorate with a mint sprig.