Вам нужно будет обернуть существующий html внутри родительского элемента mat-sidenav-container
новым mat-sidenav
для этого родителя.
Stackblitz
https://stackblitz.com/edit/angular-nure99?embed=1&file=app/sidenav-overview-example.html
HTML
<div>
<mat-sidenav-container class="sidenav-full">
<mat-sidenav #sidenav1 mode="side" position="start" role="navigation" class="sidenav-wv">
this is parent menu
</mat-sidenav>
<mat-sidenav-content>
<mat-sidenav-container class="sidenav-full">
<mat-sidenav #sidenav2 opened mode="side" position="start" role="navigation" class="sidenav-wv">
<div fxLayout="column" class="sidenav-normal">
<button (click)="sidenav2.toggle()" mat-button class="align-center color-white">
<i class="material-icons">reorder</i>
</button>
<button mat-button class="rec-menu mt-30" (click)="sidenav1.toggle()">
<span class="color-white">menu 1</span>
</button>
<button mat-button class="rec-menu mt-30" (click)="sidenav1.toggle()">
<span class="color-white">menu 2</span>
</button>
</div>
</mat-sidenav>
<mat-sidenav-content>
<mat-toolbar role="toolbar" class="main-header mat-elevation-z1 sticky-header">
<mat-toolbar-row>
<button mat-button (click)="sidenav2.toggle()" class="toggle-Mbtn new--item">
<i class="material-icons color-white">menu</i>
</button>
<h2 class="header-text header-block">This is Dashboard</h2>
<span class="flex-spacer"></span>
<button mat-icon-button>
<mat-icon class="color-white">search</mat-icon>
</button>
<button mat-icon-button [matMenuTriggerFor]="menu">
<mat-icon class="color-white">more_vert</mat-icon>
</button>
<mat-menu #menu="matMenu" [overlapTrigger]="false">
<button mat-menu-item>
<mat-icon>power_settings_new</mat-icon>
<span>Logout</span>
</button>
</mat-menu>
</mat-toolbar-row>
</mat-toolbar>
<div>
main board
</div>
</mat-sidenav-content>
</mat-sidenav-container>
</mat-sidenav-content>
</mat-sidenav-container>
</div>