Set aside 1/4 cup flour. This "bench flour" will be used to prevent sticking when rolling out later. Combine remaining flour, 1/4 tsp. salt, 3 Tbsp. olive oil, and 4 Tbsp. water in a medium mixing bowl. Mix until a shaggy ball forms, then knead and press with your hands until a firm, smooth dough forms. If dough is a bit crumbly, work in 1 Tbsp. water at a time. If dough is sticky, work in 1 Tbsp. bench flour at a time. Place dough on a piece of plastic wrap and wrap tightly. Refrigerate at least 10 minutes.

Halve and peel onions. Slice into thin strips. Stem rosemary and mince. Place a large oven-safe non-stick pan over medium heat. Add walnuts to hot pan and toast, stirring frequently, until lightly browned, 3-4 minutes. Remove from pan, set aside, and wipe pan clean.

Return pan to high heat. Add 1 Tbsp. olive oil and onions to hot pan. Cook 3 minutes. Reduce heat to medium and cook, stirring often, until onions are lightly caramelized, 4 minutes. Add butternut squash and cook, stirring often, until it begins to get tender, 5 minutes. Add brown sugar, rosemary, 1/2 tsp. salt, and 1/4 cup water. Increase heat to high, bring to a boil, and boil until liquid begins to look like a light syrup, 1-2 minutes. Remove from burner and push vegetables into center, making a 1/2" barrier around edge.

Lightly dust work surface with bench flour. Place dough on flour and lightly flour top. Using rolling pin or bottle, roll dough into a 12" circle, or same diameter as your pan. Gently place dough on top of vegetables in pan. Fold any excess dough over on itself to make the tart circular. Poke five small holes in dough to allow steam to escape. Place pan on top oven rack and bake until dough is golden brown and crisp, 13-15 minutes. Allow to rest 2 minutes.

Before flipping, make sure to protect both your hands with oven mitts or dry dish towels. Pace a large plate upside down over pan with tart. Hold pan handle with one protected hand and place your other protected hand firmly on plate's bottom. Quickly and deliberately, flip pan so plate is on bottom. Give tart gentle shake and remove from pan.

Place arugula, walnuts, goat cheese, 1/4 tsp. olive oil, and a pinch of salt and pepper into a medium mixing bowl. Toss gently to combine. Top tarte tatin with arugula salad and serve.