Melt 2 tablespoons of butter in a frying pan over medium heat. Add in one cup of rice. Allow the rice to saute in the butter for about 3 minutes, stirring constantly. Add in one cup of water and stir the rice until the whole cup of water is absorbed, which will take about 5 minutes. Then add in another cup of water. Cook and stir the rice for another 5 minutes until the water is absorbed. Lastly, stir in the last 3/4 of a cup of water. Cook and stir until that water is absorbed. The rice should be just a bit firm on the inside, but not crunchy at all. Do not overcook the rice or it will be mushy.
Take two cups of the cooked rice and place it in a bowl. (There will be a little bit of rice leftover.) Add the 1/3 cup of brown sugar and 1 tablespoon of tapioca (or corn) starch. Mix well.
In a separate bowl, combine the hot milk, pumpkin, eggs, vanilla, cinnamon and pumpkin pie spice. Whisk together until well blended. Add this mixture to the rice mixture and stir until combined.
Spray an 8x8 baking pan with cooking spray. Pour the rice pudding inside the pan and bake at 350 degrees for 30 minutes.
While the pudding is baking, mix together the cold butter, 1/2 cup brown sugar, walnuts and cinnamon for the streusel.
After the rice pudding has baked 30 minutes, sprinkle the streusel on top of the pudding and bake for an additional 30-40 minutes, or until the pudding is baked through. The center of the rice pudding should not be liquidy at, but should be firm as a pumpkin pie texture would be.