Preheat oven to 425°F (or 350°F or slightly higher for faster cooking).
Lightly oil the potatoes with olive oil and season them all around with salt. Poke them if desired (optional).
Bake the potatoes for 45 minutes to an hour, or until a cake tester goes in and out with no resistance.
While the potatoes cook, prepare the Blue Lobster: gently massage the lobster's head to calm it, or place it in the freezer briefly. Then, using a knife, quickly and precisely kill the lobster by inserting the knife between its eyeballs.
Bring a pot of water to a boil and drop the Blue Lobster into the boiling water. For a 2.2 lb lobster, cook for 8 minutes; for a 1–1.25 lb lobster, cook for 6 minutes. Halfway through cooking, rotate the lobster for even cooking.
Prepare an ice bath in a bowl: fill a bowl with ice and water, and after cooking immediately transfer the Blue Lobster to the ice bath to stop cooking and cool.
Skim any gunk from the lobster cooking water and reserve this broth for cooking the gnocchi.
Once the Blue Lobster is cool, twist off the claws. Extract the meat from the tail by crushing the shell with your wrist and pulling out the meat. For the claws, carefully crack the shell with the back of a knife to extract the meat and cartilage. Cut the lobster meat into pieces roughly the size of gnocchi and reserve some pieces for garnish.
When the potatoes are done, cut them in half and allow them to cool until they are no longer steaming hot but still warm enough to work with.
Rice the cooled potatoes onto a clean surface (like a board).
Scramble the egg (use one egg per 1–1.5 lbs of potato).
Sift a small amount of 00 Flour over the riced potatoes (start with a little, expecting to use 1/2 to 1 cup total).
Add the scrambled egg to the potatoes and gently incorporate it using a fork, without kneading.
Add more 00 Flour gradually, using a bench scraper to cut and fold the dough to expose wet bits, until the dough is soft, pliable, and not sticky — aim to use as little flour as possible.
Dust the gnocchi dough ball with 00 Flour.
Take a portion of the dough and gently roll it into a thick rope on a lightly floured surface. Do not apply a lot of pressure. Smooth the rope and push in the ends to round them.
Once all ropes are formed, dust them again with 00 Flour. Cut the ropes into gnocchi pieces and gently toss the cut gnocchi in 00 Flour to coat openings and prevent sticking. Store them on a floured surface, or refrigerate while preparing the sauce.
Finely chop the fresh Parsley.
Cut cold Butter into knobs (approx. 3–4 tablespoons).
Grate Parmigiano Reggiano to taste.
Bring the reserved lobster broth back to a boil.
In a high-rimmed saucier, add 3–4 tablespoons of Breadcrumbs and toast them dry over medium-high heat until golden brown and crispy, adding a little olive oil to coat them if desired. Transfer the toasted Breadcrumbs to a paper towel to drain, wipe out the pan, and return it to the stove (off the heat for now).
Strain any excess 00 Flour from the gnocchi and gently transfer them into the boiling lobster broth. Cook for about a minute, or until they float.
While the gnocchi cook, start the sauce in the wiped-out saucier using residual heat: add some starchy pasta water from the gnocchi cooking and squeeze the entire Lemon through a strainer into the pot. Add one of the lemon halves to the pot to extract more lemon essence.
Once the gnocchi float, gently strain them and add them to the lemon broth in the saucier.
Add the cold Butter knobs to the gnocchi and sauce. Gently stir and toss with a spatula until the butter slowly melts and emulsifies into the lemon broth. Remove the lemon half from the pot.
Once the butter is melted, add the Parmigiano Reggiano to the sauce and stir until the sauce thickens and coats the gnocchi. Add the lobster chunks (reserving some for garnish) and the chopped Parsley to the sauce and heat the lobster through.
If you want a tighter coating on the gnocchi, add a little more Parmigiano Reggiano.
To plate, warm bowls. Spoon the gnocchi and lobster into the center, distributing the lobster meat evenly. Garnish with reserved lobster chunks and the lobster claws in the center, then sprinkle with the toasted Breadcrumbs.
