If the shrimp are frozen, put them in a bowl of cold water to thaw as you prep the sauce ingredients.
Heat a skillet over medium heat and season generously with olive oil. When the oil is shimmering, add the white onion and sauté for 5-7 minutes, until soft and tender.
In the meantime, fill a large pot with water. Salt with a teaspoon of kosher salt. Taste to see if the water is comfortably salty. If it's too salty, dump some out and fill with more water. We want that perfect saltiness for our pasta!
Heat the salt water over medium high heat and bring it to a boil.
Return to the sauce. If necessary, add a little more olive oil. We don't want a dry pan. Add the garlic paste, minced parsley and cherry tomatoes and continue sauté on medium/medium low heat for 2-3 minutes.
Add the lemon juice, clam juice, white wine, and 1/2 tsp salt. Change the heat to a simmer and cover the pot. Let the mixture cook for 5 minutes, until fragrant and the tomatoes begin to soften.
Add tomato paste and balsamic vinegar. Thoroughly mix together and cover. Let this simmer, while covered, for about 5-10 minutes. Add more olive oil if the mixture seems to be getting too thick.
Your pasta water should be boiling by now. If you're using dried linguine, add it to the pot and let it cook for approximately 7-9 minutes, until al dente. If you're using fresh linguine, do this step last as it will take 1 minute for it to cook.
Meanwhile, finish prepping the shrimp. Pull the tails off and split each one, by slicing them down the middle on the outside, without cutting it in half.
Drain the pasta, reserving 3/4 cup of pasta water. Leave 1/4 cup of pasta water in the pot with the linguine, to prevent it from sticking. Save the other half for the sauce.
While stirring the sauce, slowly incorporate the pasta water into the sauce. Once half of the water is incorporated (about 1/4 cup,) stir the sauce for one minute to check the thickness. If the consistency feels like a thicker paste, add more of the pasta water and continue to stir. Turn the heat up to medium, until it starts to lightly bubble. Continue to cook and stir periodically for 5 minutes, until the water is fully incorporated and the sauce is thickened.
Add the shrimp to the sauce and let them cook for 2-3 minutes, until tender.
Remove the sauce pan from the heat and set aside.
Drain the rest of the water from the pasta and combine the linguine and sauce in a large pot. Fully coat the pasta in the sauce, with the shrimp dispersed throughout.
Plate the shrimp pasta and top generously with fresh parsley. Serve immediately.