Fill a large stockpot with 12 to 14 cups of water. Add a generous amount of salt and bring to a rolling boil over high heat. Add the chicken breasts and cook until fully cooked through (they will float to the top when done). Remove the chicken to a plate to cool, then reduce the heat to medium-low, keeping the broth at a gentle boil.

Add the butter, cream of chicken soup, and a generous pinch of cracked black pepper to the broth. Stir until the butter and soup are fully melted and incorporated into the broth.

In a medium bowl, whisk together the self-rising flour and salt. Create a well in the center and add the canola oil and egg. Use a fork to stir the mixture until a shaggy dough forms. If the dough is too dry to press together into a ball, add 1 to 2 tablespoons of cold water, one tablespoon at a time, until it holds together.

Lightly flour a work surface. Turn the dough out and divide it into three portions. Working with one portion at a time, roll it out to about 1/4 inch thickness using a floured rolling pin.

Cut the rolled dough into roughly 1-inch squares using a butter knife or pizza cutter.

Toss the dough squares in the excess flour from the work surface; this prevents sticking and helps thicken the broth. Add the squares to the simmering broth a few at a time, along with any remaining flour from the surface. Stir gently. Repeat with the remaining dough portions, re-flouring the surface as needed.

Once the chicken is cool enough to handle, shred it into bite-sized pieces and add it to the pot.

Let the dumplings simmer, stirring occasionally but gently, until they turn white and begin to float to the surface. This usually takes 20 to 30 minutes. Taste a dumpling to ensure it is cooked through and no longer tastes raw.

Adjust the seasoning with more black pepper if desired. Serve hot.