In a small saucepan, melt the butter over medium heat. Let the butter simmer gently until it begins to brown, about 4 to 5 minutes. The butter will foam and then become clear, with the milk solids dropping to the bottom of the pan. Swirl the pan and stir occasionally while the butter is cooking to make sure it browns evenly.
When the milk solids turn brown, remove from the heat and pour into a mixing bowl. Allow the butter to cool for a few minutes before dividing. Spoon off and set aside 1 tbsp of the butter to brush the tops of the finished rolls. Spoon off another 5 tbsp of the butter to use while making the dough. Be sure to catch some of the browned bits from the butter.
Heat the water to between 105°F and 115°F. Add the sugar and yeast. Stir briefly and then set aside for the yeast to activate. When the yeast is bubbly and risen, after about 5 minutes, proceed with the recipe.
In the bowl of a stand mixer fitted with a dough hook, combine the warm milk, the reserved 5 tbsp of browned butter, and 2 tbsp of honey, stirring to dissolve the honey. Add the yeast mixture and stir together. Add the flour and salt to the mixer.
Mix, using the dough hook, until the dough starts to come together. Scrape down the sides occasionally to encourage the dough to come together. If the dough is too sticky and sticks to the side without forming a ball, then add more flour, a tablespoon at a time.
Continue mixing with a dough hook on low speed for about 5 minutes until the dough is soft and stretchy. After mixing, take the dough out and lightly knead a couple of times on the counter to bring it together into a smooth ball.
Transfer the dough to a greased bowl. Rub a little oil on top of the dough. Cover with cling film or a tea towel and then set in a warm place to rise for about 1 1/2 hours until double in size.
Prepare the whipped butter by whisking it over an ice bath. Fill the bottom of a large bowl with a layer of ice cubes. Place the mixing bowl containing the remaining browned butter into the large bowl and nestle within the ice cubes. Begin whisking the melted butter while it cools. The ice cubes will chill the bowl and harden the butter as you whisk. After about 1 minute, add the remaining 3 tbsp of honey and whisk to combine.
Continue whisking the butter until it thickens to the consistency of soft, spreadable butter. Transfer to a serving bowl, cover, and set aside at room temperature until ready to serve. Refrigerate if keeping the butter overnight.
After the dough has risen, punch it down to deflate. Transfer the dough to a lightly floured counter and divide into 12 equal pieces. Shape each piece into a small ball. Spray a 10-inch cast iron pan with non-stick spray. Arrange the rolls in the pan. Alternatively, use a 9 or 10-inch cake pan. Cover with a tea towel and let rise for 30 minutes.
While the dough is rising, preheat the oven to 350°F. Bake the rolls for 22 to 25 minutes until golden brown and cooked all the way through. Rotate the pan in the oven after 15 minutes to encourage even browning.
Immediately brush the tops of the rolls with the reserved tablespoon of browned butter with a pastry brush. Let cool for 5 to 10 minutes before serving. Serve the rolls with the whipped honey browned butter and enjoy!