Combine yeast and 1½ cups warm water (about 110°) in the bowl of a stand mixer fitted with the dough hook. Let sit until foamy, about 10 minutes.
Add 4 cups flour and 1 Tbsp. salt to yeast mixture. Mix on low speed until a slightly shaggy dough forms. Pour in ½ cup oil and mix on low to incorporate. Increase speed to medium and mix until a uniform dough forms, about 6 minutes. Transfer dough to a lightly oiled large bowl and cover loosely with a kitchen towel or plastic wrap. Chill dough until doubled in size, about 18 hours.
Let dough come to room temperature, about 1 hour.
Grease a large rimmed baking sheet with 3 Tbsp. oil. Turn dough out onto a floured work surface and roll to a 16x11" rectangle. Using rolling pin, roll up dough and transfer to prepared sheet. Using your fingertips, spread dough to edges of pan. Cover dough with a kitchen towel and let rest 30 minutes.
Preheat oven to 400°. Halve squash lengthwise, scrape out seeds, and slice crosswise into ¼"-thick half-moons. Transfer to a large bowl. Add onion, thyme leaves, and 2 Tbsp. oil; season with salt and pepper.
Scatter squash mixture over dough. Drizzle oil over. Bake flatbread until crust is browned and cooked through and vegetables are light golden brown and tender, about 45 minutes. Transfer baking sheet to a wire rack and sprinkle blue cheese over while flatbread is still warm.
Just before serving, toss arugula, lemon zest, and lemon juice in a small bowl. Drizzle with oil; season with salt and pepper. Top flatbread with arugula salad.