Place the potatoes in a large pot of cold water and bring to a boil. Salt the water, lower the heat and simmer until fork-tender, about 15 minutes. Drain the potatoes and return them to the pot to dry out.
Place the vegetables in a clean kitchen towel and squeeze dry.
In a medium pot, melt the butter in the EVOO, 2 turns of the pan, over medium-low heat. Add the onion and garlic and cook until softened, 10 minutes. Add the boiled potatoes and milk and mash together. Season with pepper and the thyme, then stir in the chicken broth. Stir in the reserved vegetables and heat through, about 2 minutes. Stir in the cheese. Thin the soup with a little water if desired. Pass the crusty bread and extra cheese around the table.