Add the chicken breasts, carrots, and one of the onions, peeled and cut in half to a medium sauce pot. Cover with water and over medium heat bring to a simmer. Gently simmer the chicken for 15-20 minutes unti cooked through.
Strain the liquid and set the chicken aside to cool a bit before shredding. Place the shredded chicken in a bowl and stir in the lime juice and softened cream cheese.
Dice the second onion and mince the garlic cloves. Heat the butter in a skillet over medium heat and add the diced onions. Saute until translucent and add the minced garlic. Stir for another minute, then scrape the onion mixture into the bowl with the chicken. Stir again well. Cover the bowl and place in the fridge.
Pour the chicken broth into a large saucepan and bring it to a boil. Add the flour, bit by bit, stirring vigorously as you add it. The dough will begin to form as you stir. This takes about 2-3 minutes.
Remove the dough from the heat and transfer it to a bowl. Cover the bowl and refrigerate for 2-3 hours or overnight.
Using a small scoop portion out tablespoon-sized balls of the chicken salad onto a plate.
Next, roll out the dough. Roll a golf ball sized portion in your hand then flatten it out. Place the ball of chicken salad inside and wrap the dough around it. Pinch the end to make a drumstick shape for an authentic look! Place the balls on a tray. There should be close to two dozen.
Set up a breading station. Beat the 2 eggs in a bowl and place the breadcrumbs in another bowl. Dip the balls into the egg and then roll them in the breadcrumbs. Place them back on the tray.
To deep fry them, set a large pot of oil over medium-high heat - use just enough oil to cover the balls. When the oil is 360° carefully drop the balls in. Cook until they are golden brown then remove them using a spider strainer and place them on a paper towel-lined plate to drain.
To oven bake - heat the oven to 400 and bake for 20 - 25 minutes, moving the balls around 2 or 3 times to cook them evenly.
Enjoy warm!