Heat whole milk in a medium saucepan over medium heat. Stir until it begins to bubble and steam.
Pour 1/2 cup of the milk into the bowl of a stand mixer. Add 1/4 cup flour to the milk remaining in the pot and stir to combine into a thick paste.
Once the paste has thickened and is steaming, remove from heat and scrape into the stand mixer bowl. Then, add your cold water and eggs and stir to combine.
Add 4 cups flour, 1 tsp yeast, 1/4 cup white sugar, and 1 1/2 tsp kosher salt. Mix on low speed until the dough comes together. Then, turn the speed up to medium-low and mix for another 5 minutes.
Once the dough is smooth, turn the mixer down to low and add 14 tbsp softened butter (or brown butter) one tbsp at a time until fully incorporated. Then, mix your dough on medium speed for a few minutes until it slaps the sides of the bowl.
Remove dough from mixer and roll into a tight ball. Transfer to a greased bowl and cover tightly, proofing in a warm spot for an hour.
Rest cinnamon roll dough in the fridge overnight, for up to 16 hours. Then, remove 15 minutes before using.
Mix the filling by combining 1 cup brown sugar, 1/2 cup white sugar, 2 tbsp ground cinnamon, ground ginger, cloves, and nutmeg, and 1 tsp kosher salt.
Melt 4 tbsp unsalted butter until browned. Remove from heat and pour into your sugar and spice mixture, using a chopstick or fork to mix evenly.
Add 1 tsp AP flour to the sugar mixture and mix until combined.
Roll cinnamon roll dough on a clean work surface into a large, 24” long rectangle. Sprinkle the top generously with cinnamon sugar mixture and press lightly to adhere.
Roll the buns from the short side until you have a thick log. Then, cut the buns into 9 equal pieces.
Grease and line a 10x10” square baking pan with parchment paper. Place your buns in a grid inside, spacing about 1/2” between each bun.
Cover with plastic wrap and proof in a warm spot for 45-60 minutes until puffy and the buns are touching.
While the buns are proofing, melt 1 3/4 sticks of unsalted butter in a saucepan until browned. Then pour into a heat-proof bowl and add 1/2 tsp ground cinnamon, 2 1/2 tbsp maple syrup, and 1 tsp vanilla extract.
Let cool in the fridge for 40-60 minutes, mixing occasionally, until it’s a thick, spreadable frosting texture.
Preheat oven to 350°F. Remove plastic wrap and cover buns with tin foil. Bake for 15 minutes, then uncover and cook another 15-20 minutes until golden brown.
Remove from oven and let rest for 15-20 minutes. Remove buns from pan and finish your frosting by beating the spreadable brown butter with powdered sugar. Add 1/2 tsp kosher salt.
Spread your brown butter frosting generously over the cinnamon rolls and serve.