Melt butter in a medium pot over medium-low heat. Cook, stirring often, as butter foams and then begins to brown and smell nutty, about 5 minutes.
When golden brown, scrape into a bowl and chill until firm. Remove from refrigerator and let soften to room temperature, stirring occasionally, until spreadable. This will be split for the filling and glaze.
Combine flour, sugar, yeast and salt in the bowl of a stand mixer and whisk by hand to combine. Attach dough hook. Add milk, eggs and butter and mix on medium-low speed until a dough comes together. Increase to medium and knead until dough is smooth and dough cleans the bowl, about 15 minutes. Shape into a ball and place in a lightly greased bowl. Cover to rise for 1- 2 hours, or until doubled in size.
Transfer dough to a lightly greased work surface and roll out to a 16-inch by 12-inch rectangle. Spread the surface evenly with half the softened brown butter, leaving 1/2-inch strip along one long edge. Sprinkle with brown sugar and cinnamon. Roll the dough into a log, starting with long edge opposite the bare 1/2-inch strip. Gently pinch the edge to seal. Cut off and discard the uneven ends of the log, then slice remainder into 12 equal sized pieces.
Transfer rolls to a greased 9×13-inch baking dish (or two 9-inch round pans). Cover and let rise 45 minutes to 1 hour, or until puffy and doubled in size. Preheat oven to 350°F towards the end of the rise time. Bake rolls 20-25 minutes, or until golden brown.
Cream Cheese Glaze
Meanwhile, make the glaze. Combine softened cream cheese and remaining brown butter in a bowl, mix until completely smooth. Add powdered sugar, vanilla and a pinch of salt and stir until it comes together to a smooth, spreadable glaze.
Cinnamon rolls can be stored at room temperature, well wrapped, for several days or frozen for up to 1 month.