Whisk potato flakes, flour, and salt together in large bowl.

Whisk water and egg together in separate bowl.

Add water mixture to potato flake mixture and stir with wooden spoon until fully combined and mixture forms dough ball.

Let sit for 3 minutes for potato flakes and flour to hydrate.

Turn out dough onto lightly floured counter and knead until dough has texture of Play-Doh and springs back halfway when poked with your finger, about 3 minutes.

Lightly dust dough with flour and let rest on counter for 5 minutes.

Divide dough into 6 equal pieces.

On very lightly floured counter, roll 1 piece into ¾-inch-thick rope.

Lightly dust rope with flour.

Using floured bench scraper, cut rope crosswise into ¾-inch pieces; transfer gnocchi to lightly floured rimmed baking sheet.

Repeat with remaining dough pieces.

Press gnocchi, cut side down, on tines of fork, then roll downward to create grooves.

If dough sticks, dust your thumb and fork with flour.

Return gnocchi to sheet.

Melt butter in 12-inch skillet over medium heat.

Cook, swirling skillet often, until butter is color of milk chocolate and has toasty aroma, 3 to 5 minutes.

Add shallot and cook until fragrant, about 30 seconds.

Off heat, stir in capers and lemon juice.

Cover to keep warm.

Bring 4 quarts water to boil in large pot over high heat.

Add 1 tablespoon salt to boiling water.

Add half of gnocchi, then stir gently to keep from sticking.

Simmer until just cooked through and gnocchi float to top, about 1½ minutes.

Using spider skimmer or slotted spoon, transfer gnocchi to skillet with sauce.

Return water to boil and repeat with remaining gnocchi.

Place skillet over medium-high heat.

Cook, stirring gently, until hot throughout and gnocchi are well coated with sauce, about 2 minutes.

Season with salt to taste.

Sprinkle with Parmesan and serve.