Pat your shrimp dry with a couple layers of paper towels.
Using a sharp knife, cut the shrimp into small chunks, just larger than 1/4-inch big. Place them into a medium-sized bowl. Alternately, you could pulse the shrimp in the bowl of a food processor until you have chunks, but I tend to like the larger chunks of shrimp in my burger.
Add all of the remaining ingredients to the bowl, except the oil and buns. Use your hands or a large fork to thoroughly combine all of the ingredients. Set the bowl in the freezer for 10 minutes; this will make it easier to form the patties.
Prepare four large squares of wax (or parchment) paper that are slightly larger than your shrimp burger patty will be, by setting them on your work surface.
After 10 minutes, divide the mixture into 4 equal parts. Even though it's not hard to divide the mixture in four, I like to use my 4-ounce scoop to get the mixture out of the bowl and divide it into portions. The scoop compacts the mixture pretty perfectly and allows for minimal handling and mess.
Once you have your four portions, form them into thick patties. If you used a scoop, you just need to press down lightly on the top to flatten, and turn, smoothing the sides as you go. Transfer the squares of wax paper to a container with a lid that will fit in your fridge. Refrigerate for 2-4 hours.
Preheat a grill or a grill pan to medium. Drizzle a thin layer of oil either directly onto the top of the patties, or into a small container that you can mop generously onto the grill grates or pan. Carefully lift up the shrimp burgers, paper and all, and flip over onto the prepared hot cooking surface; peel off the paper. Close the lid of the grill, or overturn a pan on top of the grill pan. Cook for 3-1/2 minutes. Lift the lid, oil the (now) top of the shrimp burger, and gently flip it. Cover again and cook for another 3-1/2 minutes, or until firm. Remove the burgers to a plate.
Split the buns and place them cut side down onto the grill or grill pan to toast. The residual oil that is now flavored by the shrimp burgers should give the bun some nice, golden grill marks.
Serve the shrimp burgers on the toasted buns with your favorite toppings like lettuce, baby spinach, thinly sliced tomato and/or onion. I also like to mix up a mixture of mayonnaise, capers, and wholegrain mustard with a tiny shot of lemon juice to smear on the bun. You could also just serve it with a lemon wedge on the side.
It's best to eat these Shrimp Burgers immediately after they are cooked. Reheating them causes them to get "rubbery". If you do not want to serve all 4 burgers at the same time, I recommend leaving them uncooked in the refrigerator until ready to use (but use within 24 hours).