Get your pasta water boiling in a pasta pot.
Melt butter and olive oil in large sauté pan. Add mushrooms and cook for 4 minutes on Medium until they soften. Add the red pepper slices and chicken. Cook a few minutes more until pepper slices soften and chicken warms. You want some crispness still left in your peppers so this won't take more than 3 to 5 minutes. Then add the green onions. You just want them to sort of get warm as you toss them in. They don't need to really cook.
While this is happening, start your pasta if you are using dried pasta. Follow the directions on the box for the amount of pasta for 4 portions and for the timing. Angel hair usually takes about 6 minutes to cook. Fresh pasta cooks even faster.
The pasta is cooking, the ingredients above are cooking, and now your sauce will be cooking too, so you'll be keeping an eye on all three things: pasta, chicken/vegetable mixture and sauce.
Add the garlic and the red pepper flakes to your oil in a saucepan. Heat until bubbling and hot but do not over brown. Just as they start to brown on the edges, add your tomato paste, peanut butter and your hoisin sauce. Cook on Medium while you stir to mix. Then add water and continue to stir with a whisk to blend. Keep on simmering at a Low setting. Just keep warm. If you keep the temperature too hot, this sauce will continue to thicken and start to stick to your pan. If it does start to thicken too much you can add a little more water, but keep an eye on it and take it off heat if necessary. You can gently re-warm it if the pasta and other veggies aren't ready yet.
When the three parts are all cooked (pasta, chicken/vegetable mixture and sauce), drain your pasta.