score:0

write one function that does two things.

starttexthandler(e) {
    // etc
},
dosomethingelse(e) {
    // etc
},
starttexthandleranddosomethingelse(e) {
    this.starttexthandler(e);
    this.dosomethingelse(e);
}

....

onclick={this.starttexthandleranddosomethingelse}

score:0

try this:

<button onclick={()=> { this.starttexthandler(); this.secondaction(); } } text="start the test" />

score:0

you can pass an anonomus function:

<button 
    onclick={(event) => {
        this.starttexthandler(event);
        this.secondmethod(eventorwhatever)
    }} 
    text="start the test" 
/>

or a named function:

<button 
    onclick={this.handleclick} 
    text="start the test" 
/>

handleclick = (event) => {
    this.starttexthandler(event);
    this.secondmethod(eventorwhatever);
}

btw: named function give you a better stacktrace too when debugging

score:1

you could create a function that includes both events:

<button onclick={this.calltwofuncs} text="start the test" />

<script>
function calltwofuncs() {
    this.starttexthandler();
    this.sendalert();
}
...
</script>

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