Hazelnut Lace Sandwich Cookies

Pulse hazelnuts, flour, and salt in a food processor until nuts are finely ground; set aside.
Cook butter in a medium saucepan over medium heat until butter foams, then browns, about 5 minutes. Let cool about 5 minutes, then add granulated sugar, cream, maple syrup, vanilla, and 1 Tbsp. water and stir to incorporate. Return saucepan to medium heat and bring to a boil; cook, without stirring, 30 seconds. Remove pan from heat and stir in hazelnut mixture. Scrape batter into a medium heatproof bowl and let cool, stirring occasionally to reincorporate the butter.
Meanwhile, place racks in upper and lower thirds of oven; preheat to 325°F.
Scoop out rounded teaspoonfuls of batter and roll between the palms of your hands into smooth balls. Place on 2 parchment-lined baking sheets at least 3" apart. Because these cookies spread so much during baking, give each ball plenty of space. You should be able to fit 8 per sheet and will need to bake in 2 batches for a total of 32 cookies.
Bake cookies, rotating baking sheets top to bottom and front to back halfway through, until they spread into thin lacy disks and are deep golden brown around the edges, 10–14 minutes. Let cool on baking sheets. Repeat with remaining batter, using fresh parchment paper on baking sheets.
Using an electric mixer on medium speed, beat butter in a medium bowl until light, about 1 minute. Add powdered sugar and beat until pale, about 30 seconds. Add jam about 1 tablespoonful at a time, beating until smooth after each addition, then beat in lime zest.
Carefully remove cooled cookies from parchment. Using a small offset spatula, spread 1–2 tsp. filling across flat sides of half of cookies. Top with remaining cookies to create sandwiches.
Cookies can be baked 2 days ahead. Store airtight at room temperature, separated by layers of parchment paper to prevent sticking.