Dice the vegetables into similar size pieces.
Preheat the oven to 350 degrees.
Melt 2 tbsp butter in a large saucepan over medium heat. Once melted, add the vegetables and stir to coat. Cook vegetables for 20 minutes.
Meanwhile, make the white sauce by melting 4 tbsp in another, smaller saucepan over medium heat. Once the butter is melted after about a minute, whisk in the flour and salt.
Continue whisking while pouring in the milk. Whisk any lumps out.
Use the last tablespoon of butter to butter a 9x13 casserole dish (you may not use all of the butter here).
Put half of the cooked vegetables in the bottom of the casserole dish. Add half of the white sauce, then the remainder of the vegetables, then the rest of the white sauce.
Evenly spread small chunks of goat cheese over the casserole.
Bake in the heated oven for 30 minutes.
Turn the broiler on and broil for 2-3 minutes or until bubbly and brown on top.