Brown Butter:

Place the butter in a medium saucepan over medium heat. Cook until melted and continue cooking until it foams and turns golden brown and nutty, about 3-4 minutes. Transfer to a bowl and cool slightly.
Chouquettes:
Preheat the oven to 400°F (200°C). Line two baking sheets with parchment paper. Trace 1 1/4” circles on each sheet, flip so the drawing is underneath.
In a pot, combine milk, water, brown butter, salt, vanilla, and sugar. Heat until the mixture boils. Remove from heat and quickly stir in flour. Return to heat; cook while stirring for 2 minutes to dry out.
Transfer to a stand mixer with a paddle attachment. Mix on medium for 1 minute to cool, then on low, slowly add eggs until incorporated. If too stiff, add another egg.
Fill a piping bag fitted with a round tip. Pipe onto traced circles, ending each with a flick of the wrist. Flatten any points with a wet fingertip. Sprinkle heavily with Swedish pearl sugar.
Bake at 400°F for 15 minutes, then reduce to 350°F and bake another 10-15 minutes until golden. Vent each with a knife and cool on a rack.