Place pastry cream in a pastry bag fitted with a 1/4-inch plain tip; pipe cream into puffs, inserting the tip into the bottom of each. Set aside.
Prepare an ice water bath. In a small saucepan, combine 1 cup sugar, 1 tablespoon lemon juice, and 2 tablespoons water; bring to a boil over medium heat. Wash down the sides of pan with a pastry brush dipped in water to prevent crystals from forming. Continue cooking without stirring until sugar has dissolved, 5 to 6 minutes. Raise heat to high, and cook until syrup is amber-colored, about 5 minutes, swirling pan to brown evenly. Remove from heat, and dip bottom of pan in the ice bath 3 seconds to stop the cooking. Transfer pan to a heatproof surface.
Dip bottom half of each filled puff into caramel, letting excess drip back into pan. Place puffs, dipped side down, on a serving platter. Make sure the puffs are touching and adhere to one another. Once ring is complete, make another one on top of the first ring, forming a pyramid shape. If at any point the caramel begins to harden in the pan, reheat briefly over a low flame.
For the spun sugar, make a second batch of caramel with remaining 1 cup sugar, 1 tablespoon lemon juice, and 2 tablespoons water. Let cool slightly. Test by dipping a handheld whisk with the top cut off into the caramel and holding it over the pan; the caramel should fall back into pan in long golden threads. Dip whisk into caramel, and spin caramel threads over a large piece of parchment paper or onto a wooden rack. Transfer spun-sugar to croquembouche, swirling to cover.