I don't know if there is a similar agency for dogs (imagine there is) but the Meow Foundation, a cat adoption agency, has an excellent approach. After a big screening process they will match you up with a cat who fits your lifestyle. You get to foster the cat for about a two week period before you do the final "adoption". It's a good approach as it lets you confirm the cat and you work together. (I believe Iginla's cats are from here)
Another angle is to try PAWS, which is an assist dog agency. Not all of their dogs make it through all phases of the program and I believe they will look to find good homes for the ones who don't become assist dogs. I know the initial training the dogs go through and you'd end up with a very well trained dog...you'd need to go to training more than the dog would.
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