Place the venison steaks on a cutting board and tenderize with a Jaccard meat tenderizer. Pat steaks dry with paper towels. Combine 1 teaspoon each kosher salt and sugar in small bowl. Sprinkle 1 teaspoon salt and sugar mixture on 1 side of steaks and press gently to adhere. Flip steaks and repeat with the other 1 teaspoon salt and sugar mixture. Let sit at room temperature for 30 minutes.
Meanwhile, make the Caesar dressing. In a medium bowl whisk together the mayonnaise, lemon juice, anchovy paste, Dijon mustard, minced garlic, kosher salt, and black pepper. Slowly whisk in ¼ cup of the olive oil until thickened. Set aside.
Place the romaine hearts on a clean cutting board and use a sharp knife to trim off root ends, leaving bases intact so the layers stay together. Cut romaine hearts in half lengthwise. Under cool water, rinse thoroughly between layers to remove any dirt. Transfer romaine to a dish towel and pat dry. Brush olive oil on the lettuce and season all sides with kosher salt and black pepper.
Turn all gas grill burners to high; cover; and heat grill until hot, about 10 minutes. Clean and grease the grill cooking grates with tongs using a paper towel dipped in olive oil.
Use ½ teaspoon of black pepper to sprinkle on both sides of venison steaks. Important note: Steaks will now be moist but do not pat dry. Brush both sides of the steak with a little olive oil.
Set wire rack on rimmed baking sheet. Arrange venison steaks on hot grill, close lid, and cook, flipping steaks every 2 minutes and moving steaks as needed for even cooking, until meat registers 125 to 130 degrees F using a meat thermometer, about 6 minutes total. Transfer venison steaks to wire rack on rimmed baking sheet and let rest for 10 minutes.
While the steaks are resting, slice the lemons in half and sprinkle the open ends with ½ teaspoon of sugar each. Place them on the hot grill and cook until lemons have caramelized, about 6 minutes.
Meanwhile, brush romaine lettuce evenly with olive oil and sprinkle with kosher salt and black pepper. Place the romaine lettuce on the grill and cook until well browned, about 4 minutes. Flip romaine and grill until the other side is lightly charred and leaves are softened, about 2 minutes. Transfer to a serving platter.
Using a sharp knife, slice thin the venison steaks and place on top of the grilled romaine lettuce.
Drizzle with Caesar dressing and sprinkle with Parmesan, croutons, pickled red onions, and black pepper. Garnish with grilled lemons and serve immediately.