15V to 3.3V to power ESP12E

The AMS1117 is not designed for 15V input voltage. 15V is the Absolute Max voltage, while the accepted voltage range is 1.5V to 12V. Anything between the operating voltage to the absolute max is a danger zone, and anything at or above the absolute max, you are basically okay with blowing the IC up. If your voltage goes up by even a tiny bit, bad things will happen. See https://www.silabs.com/documents/public/application-notes/an1225-isodriver-amr-eos.pdf for why absolute max is not what you should be using.

Even at 15V, that's almost the limit for the output line regulation of the AMS1117-3.3 (if powered by 15V). See the data sheet, test condition for line regulation is 1.5 <= (VIN - VOUT) <= 12V.

That said, yes, you can run two voltage regulators in series. Using 15V in to AMS1117-5.0 to a AMS1117-3.3, ignoring the problem listed above, would be okay, but you are not reducing the heat problem. Linear regulators like the AMS1117 will turn regulated voltage into heat. First the AMS1117-5.0 will take (15V - 5V) * 0.35A = 3.5 Watts into heat, and the AMS1117-3.3 will take (5V - 3.3V) * 0.35A = 0.6 Watts into heat. And that's assuming it's exactly 350mA. The second one is fine, but the first one will need proper heat sinking, either on pcb copper traces, or a heatsink added to the top.

For an input voltage * current this large compared to the output, you want a switching regulator. They are much more efficient, 80 to 95%. Even at 80%, 3.3V * .35A = 1.15 Watts output means (1.15W * 120%) = 1.38W, 1.38W / 15V = 0.92A input, AND it only dissipates 1.38W - 1.15W = 0.23 Watts. You won't even feel it get warm.

There are a number of switching regulator modules you can use for cheap, or get the ics and make our own board. Some even come in simple 3 pin form factors with minimal external parts needed. Or if you just need a one off and don't mind hacking, cheap dollar store car USB adapters (the small inside the cigarette outlet ones, not the older fat ones) tend to have adjustable switching regulators you can change the resistors on to get 3.3V, and tend to accept up to 24V. (See https://hackaday.io/project/6200-quick-and-dirty-33v-power-from-5v-car-charger)