Sprinkle the cubed chicken with salt and pepper and toss & cover evenly with 3 Tablespoons flour.
Heat the oil and one Tablespoon of the butter over medium heat in a high-walled sauté pan, then add chicken, cooking until golden, about 10-15 minutes. Stir occasionally so each side of the chicken gets cooked.
Remove the chicken and set aside. Melt the second Tablespoon of butter and add the diced onions. Cook until translucent and soft. Add the garlic and remaining 3 Tablespoons of flour and mix.
Add the white wine, chicken broth, water, carrots, peas, and string beans. Cook for another 15 minutes or so.
Add in the cream, return the chicken to the pan, and mix in the seasonings. Add additional salt/pepper to taste if desired and bring to a light bubble.
Combine all dumpling ingredients thoroughly. Drop Tablespoon-sized balls into the bubbling pot.
Reduce the heat to low and cook, uncovered, for about 10 minutes, delicately turning over the dumplings so that they cook on each side.
Once the pot has thickened and the liquid has reduced, you’re ready to eat!