Thinly slice 1/2 lemon and put in a medium saucepan with 4 cups water, the garlic, bay leaf and a big pinch each of salt and pepper. Grate ½ teaspoon lemon zest and juice the remaining 1 1/2 lemons; set aside.
Bring the water to a simmer over medium heat. Add the shrimp and poach until just cooked through, 1 1/2 to 2 minutes. Transfer to a plate with a slotted spoon; reserve the saucepan and poaching liquid. Refrigerate the shrimp until chilled, 30 minutes to 1 hour.
Meanwhile, combine the peas, parsley and 1/2 cup pea shoots in a blender (preferably a high-speed blender). Add the garlic from the poaching liquid plus 1 1/4 cups poaching liquid. Carefully blend until smooth. Add 1 cup ice cubes, then add the avocados, scallions, tarragon, the reserved lemon zest and 2 tablespoons lemon juice. Blend until very smooth, thinning with cold water, 1 tablespoon at a time, if needed. Season with salt and pepper. Refrigerate at least 20 minutes or up to 1 hour.
Stir the soup and thin with cold water, if needed. Add more lemon juice, salt and pepper to taste. Divide the soup among shallow bowls.
Toss the chilled shrimp with 1 teaspoon each lemon juice and avocado oil and a pinch each of salt and pepper. Put in the middle of the soup and top with the remaining 1/4 cup pea shoots. Drizzle with the remaining 2 teaspoons avocado oil.