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First, brown the butter. All of this butter will be dispersed throughout the entire recipe.
Add the butter to a saucepan over medium heat. Melt it down and continue to stir as it goes from melted, to bubbling, to foaming. The foaming stage is when the butter begins to brown rapidly, so keep a very close eye on it. It’s ready when the butter has turned a rich amber color. (See this post all about how to brown butter for more tips).
Pour the butter into a measuring glass and scrape in the brown bits at the bottom of the pot. They hold all the flavor.
In a large mixing bowl, whisk together 2 cups of flour, sugar, salt, and yeast.
In a heat-safe glass, add the milk, water, and 1/3 cup of the browned butter (give the brown butter a stir before pouring out your 1/3 cup in order to capture some of the brown bits at the bottom of the glass).
Microwave the mixture for 30-45 seconds or until it’s warm to the touch, but not hot.
Pour the milk mixture into the bowl of dry ingredients along with the egg and vanilla. Mix to combine.
Mix in 2 more cups of flour using a wooden spoon. It should still be pretty wet and sticky, so mix in another 1 to 1 1/2 cups of flour until the dough is shaggy and tacky.
Use your hand to knead the dough into a smooth ball. It may stick to your fingers but that’s OK. But if it’s way too sticky to manage, mix in 1/4 cup of flour at a time until it’s a bit less sticky.
Cover the dough and let it rest for 10 minutes.
While it rests, make the filling. Combine all 3 ingredients in a bowl (remembering to stir the brown butter before pouring out your 1/3 cup). It should be the consistency of wet sand. Place the last bit of brown butter in the refrigerator to solidify. This will be used in the icing.
After 10 minutes, the dough shouldn’t look that different so don’t be concerned if it hasn’t puffed up. Place the dough on a lightly floured surface. Roll it into a 12×16 inch rectangle.
Cover the surface with the filling, leaving a 1/2 inch gap all around the edge of the dough.
Roll the dough up from the long end, forming one large log. Gently press the two ends in towards each other to restore the dough to its original length as it tends to stretch while rolling.
Using a piece of unflavored floss, cut off the two ends. Do so by running the floss under the log. Pull the two ends of the string up and over the log, crossing over each other. Then pull those ends through to cut through the log. Then cut the log into 12 equal pieces.
Place the pieces into a 9×13 baking dish that has been lined with parchment paper.
If you’re in a warm climate, cover the rolls and let them proof at room temperature for 45 minutes. They should double in size. If it’s cold in your kitchen, preheat the oven to the lowest temperature possible. Once preheated, turn the oven off and place the rolls inside uncovered and close the door. After 30-45 minutes, the rolls should have doubled in size.
Take the rolls out of the oven and preheat it to 350F. Bake uncovered for 25-30 minutes or until the tops are a light golden brown.
When the rolls are fresh from the oven, make the icing.
Check on the refrigerated brown butter. It should be solidified but still soft. If it’s too hard, fill a bowl with warm water and place the glass of brown butter inside for a few minutes. If it’s still liquid, continue to chill it for another 20 minutes.
There should also be 1/3 cup of butter left. If it’s just a smidge too much or too little, that’s ok. But if it’s well over 1/3 cup, take away some of the butter.
In a bowl, mix together the brown butter and cream cheese. You can do so using a fork or an electric mixer.
Mix in the powdered sugar, 1/2 cup at a time. Mix in the vanilla and milk in between each half. Then mix in the salt.
Spread the icing over the warm rolls and allow it to melt in. Now serve and enjoy!