In a Dutch oven, melt the six tablespoons of butter over medium heat. Add the onion, carrots, and celery. Cook for 5 minutes, or until the vegetables are tender.
Add the garlic and saute for an additional minute.
Evenly sprinkle in the flour, and stir to combine. Cook for 1 minute. Add the evaporated milk and chicken broth. Stir well until thoroughly combined.
Bring the milk and chicken broth mixture to a low boil, and then add the cooked chicken, thyme, pepper, and salt. Simmer uncovered for 12 minutes.
In the meantime, prepare the dumplings. In a large bowl, mix together the flour, baking powder, pepper, salt, and thyme. Add in the milk and melted butter. Use a wooden spoon to stir together until a dough forms.
Using a cookie scoop or your fingers, form the dough into golf-sized balls. Arrange the dough balls evenly on top of the broth mixture (be careful not to splash the simmering broth). Cover, and turn the heat down to a very low simmer for 15 minutes.
Check to see if the dumplings are cooked through by breaking one in half. If the dumplings are not fully done, cover again and cook for a few more minutes before testing again.
Remove from heat.
Note: If desired, you can place the Dutch oven uncovered under a broiler, to lightly brown the tops of the dumplings.