Preheat oven to 375 degrees. Pulse flour, chopped basil, 1 1/2 teaspoons salt, and the sugar in a food processor until combined. Add butter; pulse until mixture resembles coarse meal. With machine running, add cream; pulse until dough forms.
Briefly knead dough on a lightly floured surface. Divide dough into 4 equal pieces; shape into rectangles, and wrap each in plastic wrap. Refrigerate until firm, about 30 minutes.
Roll out 1 piece of dough on a lightly floured surface into a very thin rectangle (about 12 by 16 inches). Transfer to a parchment-lined baking sheet. Brush with egg wash, and sprinkle with salt. Press 10 whole basil leaves into top.
Bake until golden brown and crisp, 15 to 20 minutes. Transfer to a wire rack, and let cool. Repeat 3 times with remaining dough, egg wash, salt, and whole basil leaves. Break into individual crackers using your hands.