Combine 1-1/2 cups flour, undissolved yeast, sugar and salt in large mixer bowl. Heat milk and butter until very warm (120° to 130°F). Add to bowl with vanilla; beat 2 minutes on medium speed. Gradually add remaining 3/4 cup flour to make a stiff yet sticky dough. Fold in raspberries, blueberries and white chocolate chips. Cover bowl and let rest 10 minutes.
To prepare streusel topping, combine flour and sugar in a small bowl or food processor. Cut in butter with a pastry blender or pulse several times until crumb is the size of small peas. Set aside.
Portion dough into 12 (2-1/2 inch) greased muffin pans (do not use paper liners) using a scoop or 2 tablespoons. Sprinkle with topping. Cover and let rise 45 to 55 minutes; dough will rise but not necessarily double.
Bake in preheated 375°F oven for 20 to 25 minutes until well browned. Let cool on wire rack for 2 minutes. Run a knife around each roll to remove from pan. Serve warm.