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1. To make the streusel, combine all the ingredients except the butter, then use your fingers to rub the butter in until pea-sized or smaller chunks remain. Refrigerate while you make the bread.
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2. Heat your oven to 350F. Grease a 9X5 inch loaf pan - or an 8-inch square pan for a more coffee cake like streusel ratio - and line it with parchment paper. In a medium bowl, combine the flour with the spices, salt, baking powder, and soda.
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3. In the bowl of a standing mixer, or in a mixing bowl using an electric mixer, cream together the softened butter with the granulated and brown sugar until fluffy, stopping to scrape down the sides as necessary.
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4. Beat in the eggs one at a time and the vanilla, stopping and scraping down the sides of the bowl as needed, until fully blended together. Then beat in the applesauce - it'll look grainy.
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5. Add the dry ingredients and mix on low until just combined, then fold in the grated carrot until just combined. Scrape the batter into the prepared pan. Sprinkle the streusel evenly over the top of the batter.
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Bake the loaf until a toothpick inserted in the center comes out clean, about 45-60 minutes (less if using a square pan). Remove the loaf from the oven, allow to cool 10 minutes in the pan, then remove it from the pan (you can lift it out by the edges of the parchment) and allow it to cool completely before slicing and eating. You really do want it to be totally cooled before slicing, otherwise the texture will be off. Because this bread is so moist, it keeps, wrapped, for 5 days.
