Put all the venison ingredients in a large pot and cover with water by about 2 inches. Bring to a simmer. Taste the water, and add more salt if needed. Simmer, covered, until the meat can be easily shredded, about 2 to 4 hours.
Fish the meat out of the pot and let it cool a bit. Shred finely with two forks or your fingers. This can be done up to 3 days before you make the salad.
To make the salad, chop all the vegetables, toss with the shredded meat, and then dress with salt, olive oil and the lime juice. Serve cool.
Notes
Once shredded, the meat will keep in the fridge a week. Once the salad is made, however, it needs to be eaten that day. It will wilt overnight.