I have a mixed breed dog I adopted a year ago and i was having a lot of trouble with her chasing deer and squirrels. It hasn't ever been a huge issue but I was really hoping to one day let her off the leash for hikes, but with her history, I just couldn't take a chance.
I ended up working with a couple trainers through my school. Most of them use positive reinforcement for situations like that, and that works for a lot of people. Basically you treat them for looking at you instead of the distraction (cat, squirrel, etc). What I was finding was that as soon as I ran out of treats, our training sessions became a free for all. When first introducing the dog to our cat at my parents' house, I saw this method was reinforcing her to chase whatever it was I was trying to stop her from chasing.
One trainer suggested I look into an electronic collar as a sort of safeguard against running away if I was to start letting her off the leash. The more I researched it, the more I realized that many hunters train their dogs with these devices. So working under the guidance of a that trainer, I began getting her used to wearing the collar (I spent money on a good one with several settings). It has taken a lot of work and consistency, but I am finally seeing the results paying off. I've hardly had to use the shock function, and now I usually just use the beep function. She knows that if she hears the beep, she must return to me.
This has also worked great for training with the chickens. she actually doesn't bother them now, whether I have treats or not. I trained her with both a stop command (sort of a "freeze" command, which is awesome for situations like when a dog is running into the street or one time when she heard a loud noise and tried to run to the house before I could get to her) and I've worked really hard to get a solid recall. These two commands are really great to work on, no matter the method you use! I use the clicker with her a lot also, and she responds awesome with that.
What might work best for you is paying for a session or two with a trainer who uses methods you think will work best for your dog. It's really not too bad to pay for them (some trainers are more than others, but I really think this is worth it in the long run). I don't recommend using an electronic collar without guidance or copious research. If an owner is unsure of what they are doing, you definitely can't expect the dog to understand either. I also like to preface by saying that this training method is not for everyone, but it has worked very well for me. My dog very much enjoys her time off leash in the yard and I feel much more confident letting her off the leash. I am very consistent with her, so she is never left wondering what I expect from her.
Just ETA: I definitely agree that some dogs cannot be trained for off leash or to resist distraction and I was ready for that when I began training. I also agree that a tired dog is a happy dog and that could be an important part of a training program.
