In a high speed blender, blend the cleaned squash and water until very smooth. Strain the juice though a fine mesh strainer, taking care to not press the puréed squash through the sieve. Reserve the liquid and reserve the purée for another use like soup.
In a shallow pot on medium heat, bring the juice to a simmer for 5 minutes to remove the starchiness of the squash juice. Cool the juice in a metal bowl set over ice until very cold.
With a digital scale, weigh the juice and add water until the total liquid is about 16 oz. Note that if you have 2 cups of squash juice, there is no need to add water.
In a mixer with a dough hook, combine the juice, yeast, honey, and half the flour. Mix on low until the mixture is pancake-batter consistency. Add the salt, olive oil, and the remaining flour incrementally until all the flour is combined. Mix on medium speed for 8–10 minutes until the dough is smooth and elastic. Remove the dough from the mixer to a lightly floured surface and cover with plastic. Allow the dough to rest for 5 minutes.
Meanwhile, grease a deep baking tray with 1⁄2 cup olive oil. An aluminum turkey pan works great for this or two large Pyrex baking trays. Note: If using two smaller vessels, you will need to divide the dough in half before shaping. To shape the dough, pull the farthest edge of the dough, folding under and towards you. Rotate the dough a quarter turn and repeat the fold, under and toward you. Repeat the fold 2–3 more times, shaping the dough into a tight round ball. The surface of the dough should be taut and smooth. Place the dough in the baking tray and cover tightly with plastic wrap. Place in refrigerator and let rest overnight.
The next day, remove the dough from the refrigerator, remove the plastic wrap. Preheat the oven to 425°F (preferably with convection). With your fingertips, dimple the dough deeply and evenly. Allow the focaccia to rest 10 minutes and dimple again, spreading the dough gently with your fingertips to the corners of the tray. Repeat one more time. Cover the focaccia with a damp, lightweight towel or plastic wrap and proof in a warm area of the kitchen for 1–1 1/2 hours or until it doubles in size. Bake the focaccia for 8–10 minutes rotating the focaccia at the 7-minute mark. Remove the focaccia when the surface of the crust is deep golden brown. Cool for 15 minutes, serve warm with coarse sea salt, rosemary, and a healthy drizzle of olive oil.