Melt the butter and add the olive oil to a large, heavy-bottomed pot or Dutch oven over medium heat.
Add the onion, carrots, and celery to the pot and cook, stirring occasionally, for 8-10 minutes, or until vegetables begin to soften. Add the garlic to the vegetables and sauté for 1 minute, stirring often.
Sprinkle with the salt, pepper, and flour, stirring well to combine. Cook for 1 minute, stirring often to prevent burning.
Slowly whisk the water or chicken stock and dry white wine in with the vegetables and bring to a boil. Add the chicken, sugar, and bay leaves, cover, and reduce heat to low. Simmer for approximately 25 minutes, or until chicken is fully cooked and vegetables have softened.
Meanwhile, as the soup simmers, prepare the dumplings. In a large bowl whisk together the flour, baking powder, salt, pepper, and thyme. Use a spoon or spatula to dig a well, or hole, in the middle of the flour mixture. Add the milk and butter to the center. Use a wooden spoon to mix together into a giant dough ball. Set aside.
Once the chicken is fully cooked, remove from the pot and allow to rest until cool enough to shred. Add the peas and the chicken to the pot.
Use a large cookie scoop to form equal-sized dumplings. Add each dumpling, one at a time, directly into the simmering soup, each in a different part of the pot.
Once each dumpling has been added to the pot, gently press them down to submerge beneath the broth. Cover and reduce heat to low. Cook dumplings for approximately 20 minutes.
Once the dumplings are fully cooked, remove the cover and gently add half and half, parsley, and chopped thyme. Stir to combine.
Best served hot. Garnish with fresh chopped parsley, thyme, or hot sauce, if desired.