Make the cookies: Preheat oven to 350 degrees. Heat sugar, butter, corn syrup, and salt in a saucepan over medium heat, stirring until butter has melted and sugar has dissolved. Stir in flour and ground hazelnuts. Drop teaspoons of batter, spaced 2 inches apart, onto a baking sheet lined with a nonstick baking mat. Bake until golden, 10 to 12 minutes. Let cool. (Cookies can be stored in an airtight container for up to 2 days.)
Make the mousse: Place 2 tablespoons cold water in a bowl, and sprinkle with gelatin. Let stand for 5 minutes.
Bring sugar, corn syrup, and remaining 1/4 cup water to a boil in a saucepan over medium heat, stirring to dissolve sugar. Wash down sides of pan with a wet pastry brush to prevent sugar crystals from forming. Cook, swirling gently once mixture begins to color, until dark amber, 12 to 13 minutes.
Remove from heat. Carefully add butter, a few pieces at a time, whisking after each addition until butter melts (caramel will bubble). Add 1/4 cup each cream and creme fraiche. Add salt, and whisk until smooth. Transfer to a bowl. Add gelatin mixture, and stir until combined. Let cool completely, stirring occasionally.
In a bowl, whisk remaining 1 1/2 cups cream and 1/4 cup creme fraiche until stiff peaks form. Gently fold into caramel. Cover mousse, and refrigerate for 10 minutes (or overnight).
Assemble parfaits: Cut peel and pith from citrus. Slice fruit along membranes to release segments.
Layer segments and mousse in 6 serving glasses, beginning and ending with segments. Refrigerate, covered, until mousse is set, at least 1 hour (or overnight). Serve with lace cookies.
This dessert may be assembled up to a day before serving. Try any combination of citrus in these parfaits, preferably a mix of vibrant hues.