Peel the potatoes and cut them into 1-inch pieces. Put them into a large pot of cold water. Add a big pinch of salt and put the pot over high heat. When it comes to a boil, cook until the potatoes are just cooked through, about 20 minutes.
While the potatoes are cooking, slice the leeks into small pieces and thoroughly wash to remove any dirt. Put a medium skillet over medium-low heat and add 2 tablespoons butter. When the butter is melted, add the leeks and season with salt and pepper. Cook the leeks until they are very soft, being careful not to let them color or burn, about 20 minutes.
Drain the potatoes in a colander. Then put the potatoes back into the pot and let the potatoes dry out a bit while you finish the leeks.
In a small pot over low heat, add the milk, the remaining 4 tablespoons butter, garlic and thyme and warm it up. Strain the garlic and thyme from the warm milk and stir the milk into the potatoes using a wooden spoon. Mash with a potato masher, breaking up the big lumps. Stir in the melted leeks, taste and season with salt and pepper. Reserve 2 cups for another use if desired, such as the Round 2 Recipe Thanksgiving Pie. Cover the rest and keep warm until ready to serve.