Gain Muscle + Lose Fat = Impossible?

At the same time? No, not possible.

This why people utilize bulk and cut cycles. Eat at a small surplus and lift like a beast for a number of weeks/months, then eat at a small deficit while lifting like a beast until you cut down bf%. Repeat this until you achieve the physique you want.

Your body is not doing only one thing for long periods of time. Depending on what you recently ate, both the calories and the macros, it could be building muscle, storing fat, or burning fat, at any given time, often switching between those within a 24 hour period as it receives food, digests, and then fasts. So yes, it is possible, if you're careful about how you eat (slight deficit when not exercising, slight surplus when exercising) but it's a very slow and frustrating process prone to a lot of error and confounding issues. There are some guides with good advice on how to do a recomp.

But, cutting and bulking cycles will get you where you want to go much faster and it's usually easier.

it's complicated, but the easy/simplified answer is you can't gain muscle while at a calorie deficit.

There is such a thing as beginners gains (going from not picking up heavy things to now picking up heavy things) but this is short lived.

Now I'm big proponent of picking up heavy things while at working to lose weight because it helps you to mitigate muscle mass while eating at a calorie deficit.