In a large saucepan, cover the potatoes with cold water. Bring to a boil, salt the water and simmer for 10 minutes. Add the broccoli and cook for 8 minutes. Drain and return the vegetables to the saucepan.
While the vegetables are cooking, season the chicken with the thyme and salt and pepper. In a skillet, heat the EVOO, 2 turns of the pan, over medium-high heat. Add the chicken and cook for 6 minutes on each side.
In a small, heavy saucepan, melt the butter over medium heat. Add the flour and cook for 1 minute, then whisk in the chicken broth and 1/2 cup milk and cook until thickened, 2 to 3 minutes. Season with salt and pepper. Using a wooden spoon, stir in the cheese in a figure-eight motion until melted, 2 minutes. Remove from the heat.
Add the remaining 1/2 cup milk to the broccoli and potatoes and mash together; season with salt and pepper. 5. Cut the chicken on an angle into 1/2-inch-thick slices and place on plates. Top with the gouda gravy. Serve with the brocco-tatoes.