Combine the warm water, 2 tablespoons white sugar, and salt into the bowl of a stand mixer fitted with the dough hook. Sprinkle the yeast on top. Allow the yeast to activate for five minutes.
Once the yeast is ready, turn the speed the mixer to low and gradually add in half of the flour. Add in the egg and 2 tablespoons of softened butter, and continue to gradually add in the remaining flour. Once the flour is incorporated, turn the speed up on the mixer to medium and knead the dough for 3 minutes.
Once the dough is done, it will pull away from the sides of the bowl. Transfer it to a bowl sprayed with cooking spray and set it aside to rise for 30 minutes, allowing it to double in size.
Preheat the oven to 350 F.
Once the dough has doubled, sprinkle flour on a flat surface. Roll out the dough to ¼-inch thickness, forming a rough rectangle shape.
Spread the remaining 2 tablespoons of softened butter on the dough, and then spread the raspberry jam on top. Mix together the brown sugar, remaining ½ cup white sugar, and cinnamon, and sprinkle it over the jam.
Using your hand, roll the dough into a tight log. Cut off the uneven ends and slice the dough log into 8 rolls.
Grease a 9x13-inch baking pan with cooking spray or butter. Place the cinnamon rolls in the dish and set it aside to allow them to rise an additional 15 minutes.
After they're done rising, bake the cinnamon rolls for 16 to 18 minutes until golden brown. Once baked, remove them from the oven to allow them to slightly cool.
Whisk together the powdered sugar and milk. Drizzle the glaze over top of the cinnamon rolls before serving.