In a large bowl, combine flour, yeast, salt, and sugar.
Add 1½ cups warm water and pumpkin puree to the bowl.
Use a dough whisk or wooden spoon to mix dough until all flour is incorporated.
Cover bowl with plastic wrap and let sit at room temperature for 3 hours, or in the refrigerator for 8 to 24 hours for a slower, fermented rise.
Coat bottom and sides of a 13 x 9-inch baking pan with three tablespoons of olive oil.
Transfer dough to the pan, turning to fully coat with oil.
Gently shape dough to fit pan and cover with plastic wrap.
Let sit at room temperature until it spreads and rises to fit the pan, about 2 to 3 hours.
Heat oven to 450˚F with rack placed in the center.
Drizzle another tablespoon or two of oil over the dough.
With oiled fingers, press straight down into the dough to create deep dimples, then sprinkle with sea salt.
Transfer to the oven, and reduce heat to 425˚F.
Bake for 22 to 28 minutes until the top is golden and the bottom is crispy.
Remove from the oven and cool until slightly warm, about 15 minutes.
While bread is cooling, combine butter, garlic, and chopped rosemary.
Drizzle this mixture over the focaccia, cut it into slices, and serve.