Combine 1 cup flour, dry yeast, sugar and salt in a large mixer bowl and stir until blended. Combine water and oil in a microwave-safe bowl. Microwave on HIGH in 15 second increments until very warm but not hot to the touch (120° to 130°F). Add to flour mixture with the egg and vanilla.
Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Add remaining 1 cup flour. Beat 2 minutes at medium speed of mixer, scraping bowl occasionally.
Using a rubber spatula, spread dough in greased 8 x 8-inch pan. (Don’t worry if the dough isn’t even or filling the corners.) Cover with towel; let rise in warm place until doubled in size, about 30 to 45 minutes .
While dough is rising, prepare Streusel Topping by mixing flour, sugar, brown sugar and cinnamon in a small bowl. Cut in butter with a pastry blender or two forks until very crumbly but with small pieces of butter still visible.
Sprinkle topping over coffee cake. Bake in preheated 350°F oven for 35 to 45 minutes or until golden brown. Cool slightly on wire rack.
For frosting, combine powdered sugar, 1 tablespoon milk and vanilla in a small bowl. Add an additional tablespoon of milk if needed, to get a slightly thin frosting. Drizzle over coffee cake.