Cut the butter into cubes and place in a medium saucepan (the pan should be big enough to prevent boil overs, as the butter will foam up a lot). Place over medium heat until butter melts, shaking occasionally. Once the large bubbles subside, watch closely until the butter is golden and the solids are nicely browned. Immediately pour the melted butter into a bowl. (If the solids get a little too dark, you can strain them out and discard them.).
Rinse out and dry the saucepan. Put half the sugar in the pan, and half into a blender. Add the milk to the saucepan, and place over low heat, stirring well to dissolve the sugar.
Meanwhile, add the egg yolks to the sugar in the blender. Process until combined, then, while still running, drizzle the brown butter into the yolk-sugar mixture.
When the milk-sugar mixture is almost simmering, turn the blender on again and drizzle in the liquid. Add the salt and process until well blended. Continue to process while slowly adding the ice-cold heavy cream. Transfer to a bowl, cover tightly with plastic wrap and refrigerate until very cold, at least 2 hours or up to overnight.
Churn in your ice cream maker according to manufacturer's instructions. At the very end, two or three minutes before turning off the machine, add the fleur de sel and churn until incorporated.