Make the potatoes: Bring a large saucepan of salted water to boil. Add potatoes and cook until tender. Drain and set aside.
Make the soft boiled eggs: Meanwhile, place eggs in a medium-sized saucepan. Cover with cold water by about 2 inches. Bring water to a boil over high heat, turn off heat, cover, and let stand for 5-6 minutes. Drain and run eggs under cold water until they cool. Peel and set aside.
Pickle the cucumbers: In a small bowl, mix together ½ teaspoon Kosher salt and vinegar until salt dissolves. Add in cucumbers and toss to coat. Set aside.
Cook the shrimp: Heat olive oil in a skillet over medium high heat. Add shrimp and cook until barely pink. Season with salt and pepper to taste. Set aside.
Make the dressing: In a medium sized bowl, whisk together egg yolk, mustard, salt, and lemon juice. Slowly add olive oil, whisking continuously. Dressing should start to thicken and emulsify. Once you've used up all the oil, if dressing is too thick, add a drop or two of cold water and whisk to loosen. Stir in shallot and capers into dressing and set aside.
Coarsely chop the cooked potatoes and quarter the peeled eggs. Drain cucumbers. If desired, chop shrimp into smaller pieces.
Place quartered Romaine pieces on individual plates. Split cucumbers, potatoes, radishes, and avocado evenly amongst plates. Top with shrimp. Drizzle dressing on top of everything, leaving some to serve on the side. Add freshly ground pepper to taste.
Alternatively, you can mix everything together in a large serving bowl with chopped Romaine leaves and toss in dressing before serving.