Combine the plums, vinegar, and a pinch of salt in a big bowl.
Stir together the poppy seeds and salt on a rimmed plate. Halve the chicken breasts horizontally (so you end up with four pieces). Place parchment on top and pound the chicken to an even 1/4-inch thickness. Dredge the chicken in the poppy seeds, pressing firmly to fully coat.
Add oil to a large cast-iron skillet—you want a thin layer covering the bottom—then set on the stove over medium-high heat. When the oil is hot, add the poppy-coated chicken. (If any poppy seeds fall off mid-transfer, you can sprinkle more on.) Depending on the size of your pan, you’ll probably have to do this in batches. If the chicken is overcrowded, it’ll steam instead of brown. Cook for 2 to 3 minutes per side, until the outside is crisp and the inside is cooked through (165°F). Transfer to a cooling rack to drain and sprinkle with a big pinch of salt. If you’re cooking in batches, cook the rest of the chicken in the same way, adding more oil to the pan as needed.
While the last batch of chicken is cooking, roughly smash the soaking plum pieces with a potato masher—turning some into mush, leaving others chunky. Stir in the oil. Add the lettuce, toss, and season to taste with salt and vinegar.
Serve the poppy chicken immediately with the salad on top or alongside.