For the application that you have mentioned, the noisy relay appears to be your best friend.
Any solid state device will have a voltage drop across it when conducting. For triacs, it will be approx 1V, which when multiplied with your current gives a power dissipation of:
P = 1V x 11.4A (assuming 220V power line) = 11.4 watts
(For 110V systems, it will double up to 22.8V)
Now you have a headache of removing 11.4 watts of heating from your triac or it will get fried within seconds.
To achieve this, you will need heat sinks (big ones) and maybe a cooling fan depending upon what exact parts you select. Possible to do if you are OK with going through all this trouble. (I personally won't.)
If you are OK with a coil relay, read further:
Relay selection : You are lucky to have a purely resistive load. It makes things rather simple. Since calculated current is 11.4 A, you can go with 220 V 16 A or higher rated relay provided it comes from a good company.
Here is a circuit that you can use:
The zener can be replaced by a common diode. I have been using 1N4148 in my circuits. +9V will depend upon the coil voltage of your relay. If it's a 5V coil relay, make that +5V or you will damage the relay. Transistor depends upon the coil current. Coil current shouldn't be more than 100 mA and any general NPN transistor will handle that much current. Input signal will come from esp8266 GPIO.
For your case, you can select 12 V coil relay and a 3.3 V regulator. Power up the circuit using 12 V wall adapter. 12 V will get regulated to 3.3 V by the regulator and it can be used by esp and sensor.
Special care about sensor:
You don't want to keep the sensor enclosed in a casing. It needs to be in contact with flowing air inside room so as to give you a correct temperature value. Place it accordingly. Also, make sure its away from any components that get hot - your voltage regulator for example.
Now as far as the noise is concerned, you wont be switching your heater on/off every minute. It will happen once in maybe 10 mins in the very worst case. If you are still worried about the noise, place the circuit in a casing and it will reduce the noise to certain extent. Make sure you put the temperature sensor out of the casing. From my personal experience, noise shouldn't be a problem at all but again I am a person who doesn't wake up to his alarms so there is that.