Melting the butter and mixing it with milk or other liquid ingredients is almost always done as part of the muffin method, where a muffin, cake or quick bread batter is formed by mixing dry ingredients together, wet ingredients together, and then quickly combining the two.
In practice, the butter is not going to mix with the milk.
It is going to mostly float on top, and stay liquid.
You will have sufficient time to combine the wet and dry mixes, and create the batter.
Putting them together just makes it easier to form the final batter.
It would work just as well if you made a dry mix, then added the remaining liquid ingredients (usually milk, eggs, and maybe some flavoring), and then the melted butter.
When you stir the dry and wet mixes together, the butter is going to become fairly uniformly mixed into the batter.
Note that many quick breads are made with oil, which is treated the exact same way, even though melting is not an issue.
The only reason to use butter is because the flavor is desired for whatever item is being made.
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There are actually some recipes I've found that specifically call for adding melted butter to ice cold milk or buttermilk, intentionally creating little butter globs.
For example, in this America's Test Kitchen recipe for "Lighter Chicken and Dumplings" they do this.
FOR THE DUMPLINGS Whisk flour, baking soda, sugar, and salt in large bowl.
Combine buttermilk and melted butter in medium bowl, stirring until butter forms small clumps; whisk in egg white.
Add buttermilk mixture to dry ingredients and stir with rubber spatula until just incorporated and batter pulls away from sides of bowl.
If you watch the video , the reasoning is explained (starting at 3:20): These little clumps of butter, once they are in the dough, they are going to melt in the dumplings and form little bursts of steam that are going to help the dumplings rise.
Now, while this may be an unusual method, the results will be similar in other applications.
Because butter is partially water, when that water heats up, it turns into steam, which expands and creates airy space in the dough.
With the dumplings, the effect is a nice, light dumpling that floats (as you would hope).
In general, however, as rumtscho pointed out in a comment, all ingredients should be room temperature unless otherwise instructed, so even after melting your butter, you will need to allow it to cool and allow your eggs and milk to come to room temperature.
Doing so will prevent butter from forming clumps in recipes when they are unwanted.
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Egg yolk or citric acid are emulisifying agents readily available and oftentimes used in cooking.
I haven't experimented with these.
However, you may add a small (around 1 teaspoon per 200g smelted butter) amount of flour to the liquid butter, this prevents the butter from stagnating in large globules.
Add the flour to the butter and mix in promptly to avoid clumping of flour.
Then, add the Butter(+flour)-mixture to the liquid batter.
Be careful with adding too much flour as it may reduce the butter's temperature below stagnation before mixing it into the liquid batter.
Source: Experience and engineering degree in physics with minor in polymer technology (which includes knowledge on emulsions).
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You can whisk the eggs, add melted butter, whisk again to emulsify, then add the milk, it will all be emulsified.
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When I make pancakes, which has the same problem of mixing eggs, (butter)milk, and melted butter - what I usually do is separate the eggs, mix the butter with the yolks, and mix the whites with the milk - and only then whisk everything together.
The emulsifiers in the egg yolk help keep everything mixed together, rather than the butter just solidifying and separating out.
cooking.stackexchange.com/users/32573/Chris Macksey
I've seen a lot of recipes, particularly for things like quick breads, that call for combining melted butter with milk and eggs before mixing with dry ingredients.  Of course, if the milk and eggs are cool the melted butter will immediately solidify. Is there a technique for this that recipe writers are assuming I know? Or is it just intended that there be little droplets of solid butter suspended in the batter or dough?
Adding melted butter to cold milk in baking
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<baking><butter>
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Some recipes intentionally create these little butter globs to improve the texture, but this effect can be avoided by whisking the eggs (or yolks) with the butter first or adding a little flour to the liquid butter before adding it. When the recipe does not call for butter globules to be formed, you can avoid this happening by mixing a small amount of flour with the butter or whisking the butter with eggs before adding other ingredients.
Some recipes actually call for globules of butter to be formed. There are various methods for preventing butter globules forming, including adding flour to the butter and whisking with eggs.