1/2 cup (3 1/2 ounces) sugar
1/4 cup water
1 teaspoon lemon juice
1 cup blanched almonds, toasted
1/2 teaspoon salt
1 tablespoon vegetable oil

Line rimmed baking sheet with parchment paper; spray parchment with vegetable oil spray and set aside. Bring sugar, water, and lemon juice to boil in medium saucepan over medium heat, stirring once or twice to dissolve sugar. Cook without stirring until syrup is golden brown, 10 to 15 minutes. Remove saucepan from heat, stir in nuts, and immediately pour mixture onto prepared baking sheet. Place baking sheet on wire rack and allow caramel to harden, about 30 minutes. 

Break hardened caramel into 1- to 2-inch pieces; process pieces in food processor until finely ground, about 30 seconds. Add salt and vegetable oil and continue to process until uniform paste is formed, 1 to 2 minutes. Transfer mixture to bowl, cover with plastic wrap, and set aside. 

Pastry Dough  

3 large eggs
6 tablespoons unsalted butter , cut into 12 pieces
1/3 cup whole milk
1/3 cup water
2 teaspoons sugar
1/2 teaspoon salt
3/4 cup (3 3/4 ounces) all-purpose flour
2 tablespoons toasted, slivered almonds

Adjust oven racks to upper-middle and lower-middle positions and heat oven to 400 degrees. Draw or trace 8-inch circle in center of two 12 by 18-inch sheets of parchment paper; flip parchment over. Spray 2 baking sheets with vegetable oil spray and line with parchment (keeping guide rings on underside).

Beat eggs in measuring cup or small bowl; you should have 2/3 cup (discard excess). Heat butter, milk, water, sugar, and salt in medium saucepan over medium heat, stirring occasionally. When mixture reaches full boil (butter should be fully melted), immediately remove saucepan from heat and stir in flour with heatproof spatula or wooden spoon until combined and no mixture remains on sides of pan. Return saucepan to low heat and cook, stirring constantly, using smearing motion, until mixture is slightly shiny and tiny beads of fat appear on bottom of saucepan, about 3 minutes.
Immediately transfer mixture to food processor and process with feed tube open for 30 seconds to cool slightly. With machine running, gradually add eggs in steady stream. When all eggs have been added, scrape down sides of bowl, then process for 30 seconds until smooth, thick, sticky paste forms. 
Transfer ¾ cup dough to pastry bag fitted with 3/8-inch round tip. To make narrow inner ring, pipe single ½-inch-wide circle of dough directly on traced guide ring on 1 baking sheet. For large outer ring, squeeze out any excess dough in pastry bag and change pastry bag tip to ½-inch star tip. Put all remaining dough into pastry bag. Pipe ½-inch-wide circle of dough around inside of traced guide ring on remaining baking sheet. Pipe second ½-inch circle of dough around first so they overlap slightly. Pipe third ½-inch circle on top of other 2 circles directly over seam. Sprinkle chopped nuts evenly over surface of ring.

Place sheet with larger outer ring on upper rack and sheet with narrow inner ring on lower rack and bake until narrow ring is golden brown and firm, 22 to 26 minutes. Remove narrow ring and transfer to wire rack. Reduce oven temperature to 350 degrees and continue to bake larger ring for another 10 minutes. Remove baking sheet from oven and turn oven off. Using paring knife, cut 4 equally spaced 3/4-inch-wide slits around edges of larger ring to release steam. Return larger ring to oven and prop oven door open with handle of wooden spoon. Let ring stand in oven until exterior is crisp, about 45 minutes. Transfer ring to wire rack to cool, about 15 minutes.