In a small saucepan, brown butter: Melt butter over medium heat until it begins to foam and sputter, swirling the pan occasionally. Continue to heat and swirl until it begins to smell nutty, its color darkens, and small gold bits form on the bottom of the saucepan. Remove from heat immediately and set aside to cool slightly.
In a bowl of a standing mixer fitted with the paddle attachment or a large mixing bowl, combine 1 and 1/2 cups all purpose flour, whole wheat pastry flour, brown sugar, granulated sugar, yeast, salt, cinnamon, nutmeg, ginger, and cardamom.
With a small whisk, stir the browned butter to scrape up any brown bits, then pour into the dry ingredients. Add the warm milk, pumpkin, egg, and vanilla. Mix on medium speed (or by hand with a wooden spoon) just until the dough comes together.
Switch to a dough hook and mix for 5 minutes on low, stopping periodically to scrape down the bowl (or knead by hand for 7 minutes). The dough will be wet and sticky. Continue to knead, gradually working in an additional 1/3 to 1/2 cup all purpose flour. The dough should lose some of its stickiness but still feel quite tacky.
Lightly spray a large mixing bowl with cooking spray. Place the dough into the bowl, turning once to coat it with oil, then cover the bowl with plastic wrap or drape a kitchen towel over the top. Let rise in a warm, draft-free place for 1 to 1 1/2 hours, until doubled in size.*
Meanwhile, make the filling. In a small saucepan, brown the 4 tablespoons butter, following the same procedure for Step 1 above. Set aside. In a separate small bowl, whisk together the brown sugar, granulated sugar, cinnamon, nutmeg, and salt. Grease a 9x5x3 inch loaf pan.
Turn the risen dough out onto a well-floured surface and sprinkle a bit of additional flour on top. With a rolling pin, roll the dough into a 12-inch by 20-inch rectangle so that the long side is nearest to you (take care to measure with a ruler). With a pastry brush, spread the browned butter over the dough, all the way up to the edge. Sprinkle with the cinnamon-sugar mixture (it will seem like a lot, but use all of it).
Slice the dough vertically, into five strips of equal width. Carefully stack the strips on top of one another (a long spatula is helpful) so that you have a single stack of five strips. Slice the stack into five equal slices again, so that you are left with five stacks of five squares. Line up each stack of dough squares in the pan so that the cinnamon sides are facing the narrow end of the pan. The slices will be longer on one side—stand them on their shorter end so that each slice is taller than it is wide. Place a kitchen towel over the loaf pan and allow it to rest in a warm, draft-free place for 30 to 45 minutes or until almost doubled in size.
Preheat to 350 degrees F. Bake loaf for 25 minutes, tent with foil, then bake an additional 20-25 minutes, until the top is a deep golden brown and the internal temperature of the dough slices registers 190 degrees F. (If the top is only lightly brown, the dough may not be done in the center.)
Remove loaf from the oven and allow it to rest in the pan for 20 to 30 minutes on a wire rack. Run a butter knife around the edges of the pan to loosen the bread and invert it onto a cutting board. Serve warm.
Notes
*To determine if the dough has risen sufficiently, press the dough lightly with two fingers. If the indentation remains, the dough is fully proofed and ready to use.
TO MAKE AHEAD: After kneading, place the dough in the refrigerator and let rise overnight. When you are ready to continue, let the dough rest at room temperature for 30 minutes, then proceed with the recipe as directed.
TO STORE: Keep leftover pumpkin pull apart bread well-wrapped in plastic or foil at room temperature for up to 2 days.