Make the dough: Mix 1 1/2 cups flour, the milk and yeast in a stand mixer fitted with the paddle attachment on low speed until combined, about 45 seconds.
Cover the yeast mixture with the remaining 1 1/2 cups flour, the sugar and salt; set aside until the yeast mixture starts to rise through the flour and the mixture looks craggy, about 45 minutes. Add the eggs and mix on low speed until just incorporated. Increase the speed to medium high and beat until the dough is smooth, uniform and starts to pull away from the sides of the bowl, about 5 minutes.
With the mixer running, add the 3 sticks butter, 1 cube at a time, making sure each piece is incorporated before adding the next. Scrape down the sides of the bowl and mix 1 more minute. Transfer the dough to a separate large bowl, cover with plastic wrap and refrigerate 8 hours or overnight.
Brush a 10- to 15-cup Bundt pan with the melted butter to coat. Place the dough in the pan, gently patting it down until it is even. Cover with plastic wrap and let rise in a warm spot until the dough fills two-thirds of the pan, 2 to 2 1/2 hours.
Position a rack in the lower third of the oven and preheat to 375 degrees F. Bake the cake until dark golden brown, about 35 minutes. Transfer to a rack and let cool 15 minutes in the pan, then invert onto the rack to cool completely, about 2 hours.
Make the syrup and filling: Combine 1/4 cup each sugar, raspberry jam and boiling water in a bowl, stirring to dissolve the sugar; set aside. Invert the cake again and make eight 2-inch-deep, 1-inch-wide incisions evenly spaced around the bottom of the cake with a paring knife. Fill a pastry bag fitted with a 3/4-inch tip with the remaining 1 1/4 cups jam and pipe it evenly into the incisions. Let the cake sit 5 minutes, then invert back onto a plate and brush with the raspberry syrup. Let sit 5 minutes, then sprinkle with the remaining 1/4 cup sugar.