Place the potatoes in a large pot and cover with cold water. Bring to a boil, salt the water, reduce the heat a bit, and cook until tender, about 12 minutes. Heads up: about halfway through you will add the broccoli to the pot, so use a pot large enough to accommodate both.
Season the chicken breasts with the thyme, salt, and pepper. Heat the EVOO in a skillet over medium-high heat and cook the chicken for 6 minutes on each side.
Add the broccoli to the potatoes and cook for 6 minutes.
Heat the butter in a small saucepot over medium heat. When the butter has melted, add the flour and cook for 1 minute, then whisk in the chicken stock and 1 cup of the milk or half-and-half. Thicken for 2 to 3 minutes, season the sauce with salt and pepper, then stir in the cheese in a figure-eight motion with a wooden spoon for 2 minutes, or until the cheese is melted. Turn off the heat.
Drain the cooked broccoli and potatoes and return to the hot pot to dry them out. Mash the broccoli and potatoes together, using the remaining milk to reach the desired consistency. Season with salt and pepper.
Slice the chicken breasts on an angle. Pile some mashed broc-o-tatoes on plates and arrange the fanned-out chicken on the edge of the mound. Pour the Gouda gravy over the top.