Preheat the oven to 350 F. To make the dough, combine the butter, oil, and brown sugar. Mix for a couple of minutes until it is smooth. The mixture should be lighter in color and fluffy. Add the egg and vanilla and mix again. Add the remaining ingredients and mix until just combined.
To make the streusel, combine all the ingredients in a bowl and mix with a spoon or your hand. Start with 1/3 cup of flour. The mixture should stick together, but should also crumble when you run it through your fingers. If it isn't crumbling, add 1 tbsp additions of flour until it crumbles.
Using a large (1/4 cup) cookie scoop, scoop a ball of dough onto a lined baking sheet. Using the back of your cookie scoop or your fingers, make an indent on the tops of the dough. Sprinkle a couple of tablespoons of the streusel into the indents on the tops of each cookie.
Bake the cookies at 350 for 10-12 minutes. While they are baking, make the icing. Combine the powdered sugar, milk, and vanilla in a small bowl and whisk until it's smooth and runny. Once the cookies have cooled slightly, drizzle on lots of icing! Enjoy!