In a separate sauce pot, add the salted butter and all-purpose flour.
Increase heat to medium while stirring.
Continue to stir butter and flour mixture together with a rubber spatula, and cook until the mixture smells like toasted flour and begins to turn a very pale golden color, or about 8-10 minutes.
Slowly whisk in the remaining chicken stock into your butter and flour mixture, making sure that all of the liquid is fully incorporated into the butter and flour.
Cook the sauce for roughly 5-10 minutes on medium heat, stirring continuously.
Add the tiny splash of sherry vinegar to the sauce.
Taste the sauce and adjust for salt and pepper until it tastes fantastic.
Remove the sauce from the heat and set aside for later.
In a separate soup pot, add the cut russet potato chunks and cover by 2” with cold water.
Bring the potatoes to a simmer, and add kosher salt until the water tastes pleasantly salty.
Boil the potatoes for roughly 20 minutes, or until they are completely tender.
Drain potatoes in a colander, and return them to the pot.
Mash the potatoes with a fork or potato masher, and quickly stir in the butter and warm heavy cream until fully incorporated.
Season the mashed potatoes with kosher salt and freshly-ground black pepper.
Set the mashed potatoes aside for later.
Bring a medium-sized pot of salted water to a boil.
Add the green beans and cook until just barely tender, or about 4-6 minutes.
Drain green beans and add the 1 Tbsp salted butter. Mix until the butter has evenly coated the cooked beans.
Taste and season green beans with a sprinkle of salt & freshly-ground black pepper, if desired.
Add the cooked diced white onions and cooked mushrooms from earlier to the velouté sauce, and bring to a gentle simmer.
Add the chicken breasts to the velouté sauce and vegetables, and warm them up until the chicken breasts are cooked, or until they reach an internal temperature of 160F.
Serve plates up with the creamy mashed potatoes, onions, mushrooms, chicken breast, velouté sauce, and green beans.
Garnish the plates with the finely-sliced fresh parsley and fresh chives.