Add your walnuts to a large (dry) pan over medium heat and toast until they begin to lightly char, shaking frequently. Remove from pan and crush just before serving (either with your hands, the bottom of a mug, in a zip lock bag etc).
In the same pan add a good drizzle of olive oil (around 1 tbsp) and add your butternut squash. Fry until it begins to brown, stirring every now and then. Season with a good pinch of salt and pepper, then continue frying until golden and crispy. Remove from pan.
In the same pan add your mushroom slices (and more oil if needed) and begin frying until they begin to wilt. Season with a pinch of salt and pepper, then continue frying until they begin to brown.
At this point pop your pasta in heavily salted boiling water and cook until al dente. Don't pour in a tonne of water, only enough to efficiently cook the pasta. You want it fairly shallow so it's extra starchy. Just before draining, scoop out a cup of the salty starchy water.
Melt butter into the pan with the mushrooms. It will begin to foam, just keep gently stirring until the butter starts turning from a lemony yellow to a light brown. After a few more minutes you will start to see tiny dark specks start to form on the surface, these are just the milk solids. Add in your garlic and thyme leaves, give it a stir and continue browning for another 1-2mins. Once you see the butter turning a deeper brown colour, it will release a nutty aroma. This is when your butter is properly browned. If at any point the nutty aroma smells more like burning, then simply reduce the heat.
Pour in 1 cup starchy pasta water and quickly whisk to emulsify the water with the butter. It should start turning into a browny coloured sauce. Once it starts to thicken up stir in your pasta and butternut squash. Sprinkle over goats cheese, give the pan ONE toss (don't stir or the cheese will melt and go gloopy - alternatively you can serve cheese sprinkled over individual portions).
Serve by squeezing over lemon juice (to taste) and a good sprinkling of crushed walnuts.