Bring a large covered pot of salted water to a boil. Fill a large bowl about 3/4 of the way full with ice water.

Season shrimp on both sides with kosher salt and black pepper.

Set out three medium-sized bowls. To the first bowl add the cornstarch and flour and mix to combine. To the second bowl, add eggs and whisk until combined. To the third bowl, add panko breadcrumbs. Working in batches, coat shrimp individually in cornstarch and flour mixture. Shake to remove excess. Transfer shrimp to egg mixture and coat completely. Shake off excess egg and transfer to panko dish. Cover the shrimp with panko and press down into the shrimp, ensuring all sides are coated. Transfer shrimp to a plate and set aside.

When water comes to a boil, add broccolini and cook for two minutes. Remove broccolini from water and plunge into the ice water bowl, submerging for about 1-2 minutes. Leave the pot of boiling water on the stove. When broccolini is cooled, remove from ice water and slice on a diagonal into about 1" pieces. Set aside.

Heat 1/4 cup olive oil in a large dutch oven over medium to medium-high heat. Working in batches, cook the shrimp for about 1-2 minutes per side, until the coating is crispy and golden brown. Remove to a paper towel-lined plate (and if possible, transfer plate to a low heat oven where it can stay warm). Repeat with remaining shrimp. When all the shrimp are cooked, there may be browned or burned parts of the coating remaining in the oil. Carefully use a couple paper towels to wipe out any excess oil and burnt bits. Return dutch oven to the stove and turn heat down to medium to medium-low.

Add linguine to pot of boiling water and cook until al dente. Drain pasta, reserving about 1/2 cup pasta water.

While pasta is cooking, melt butter with remaining two tablespoons olive oil in the dutch oven. Add shallot and broccolini and sauté, stirring occasionally, for about 4-5 minutes. Add garlic and cook for one more minute, stirring occasionally. Add whit wine and cook until it has reduced by about half. Add lemon zest, lemon juice, and heavy cream. Stir to coat vegetables and allow to cook for about 4-6 minutes, stirring occasionally. Season with salt and pepper, then add cooked pasta directly to the pot. Stir until pasta is coated with sauce evenly, adding a splash of pasta water if needed. Serve topped with cooked shrimp.