Preheat oven to 325 degrees F. Season beaver hindquarters all over with kosher salt.
In a small bowl, combine celery salt, black pepper, onion powder, garlic powder, smoked paprika, and cumin. Set dry spice mix aside for later.
Heat oil in a 6-quart Dutch oven on stovetop burner over medium-high heat. Add beaver hindquarters and cook for about 3 minutes on one side. Flip over and cook another 3 minutes on the other side.
Add onion and chicken broth, and ONLY HALF of the dry spice ingredients to the Dutch oven. Save the other half of the spice mix to add toward the end of the cooking time. Mix well, cover with lid or tightly with foil, and place in the oven at 325 degrees F, undisturbed, for 4 hours.
Take Dutch oven out, remove lid, and use tongs or forks to shred meat off of the bones. Discard the bones.
Preheat oven to 425 degrees F.
Drain off any remaining liquid in the Dutch oven and set liquid aside to use as a broth base for your next soup.
Lay Texas toast slices on a baking pan and place on middle shelf of the oven. Bake for about 7 minutes.
Meanwhile, add remaining half of the spice mix and the queso to the beaver meat and mix well. Simmer on stovetop, without the lid, over medium heat for about 5 minutes while the Texas toast is baking.
Serve cheesy beaver meat mixture on the warm Texas toast with lots of hamburger dill pickles.