Set a Dutch oven over medium-high heat. Add olive oil. Pat dry chicken breasts and sprinkle with salt and pepper on both sides. Once oil is hot, add chicken and cook for about 4 minutes per side. The chicken may still be pink in the center - that's okay. They will finished cooking later. Let cool for 10 minutes and then dice into 1-inch chunks.
Reduce heat to medium and add butter. Once melted, add diced onion, carrots, and celery. Cook for about 5 minutes. Add seasonings, minced garlic, Worcestershire sauce, and hot sauce. Cook for 1 more minute. Add flour and stir together to make the roux. Cook for 2 minutes.
Add chicken broth about ¼ cup at a time to roux to create a thick base. But make sure to pause, stir, and bring mixture back to a simmer before each addition. Do the same with the half and half. This ensures the mixture stays thick. Add in the frozen peas and chicken to the mixture and stir together.
Make the dumplings by first mixing the dumpling dry ingredients together in a large bowl. Heat milk and butter together until the butter melts. Add to the dry igredients, along with the sour cream. Mix together with a wooden spoon just until combined. The mixture should be thick and hard to stir once it's ready.
Using a cookie scoop or ice cream scoop, scoop the dough and drop gently into the soup. Repeat until all the dough is gone, making sure the dumplings are about 2 inches apart. Spoon a little soup over each dumpling. Reduce temperature to medium-low and cover for 15 minutes. Do not remove lid at any time to prevent steam from escaping. This steam will cook the dumplings.
Use a toothpick to check the doneness of the dumplings. If set, turn off the heat and let cool for about 10 minutes. This will ensure the soup is cool enough to eat. Serve in a bowl by scooping up a dumpling or two and covering with the soup. Enjoy!