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Let's see if now I got it better...

DynamicModule[{var = False},
    
Dynamic@Column[{
      PopupMenu[
        
Dynamic[var], {False -> "one menu"#, 
        True -> "Add another Menu"}],
       
If[var, #0[]#0["new menu"], ## &[]]}]] &[]&["one menu"]

Let's see if now I got it better...

DynamicModule[{var = False},
   Dynamic@Column[{
      PopupMenu[
       Dynamic[var], {False -> "one menu", 
        True -> "Add another Menu"}],
      If[var, #0[], ## &[]]}]] &[]

Let's see if now I got it better...

DynamicModule[{var = False},  
Dynamic@Column[{PopupMenu[  
Dynamic[var], {False -> #, 
True -> "Add another Menu"}],    
If[var, #0["new menu"], ## &[]]}]] &["one menu"]
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Rojo
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Let's see if now I got it better...

DynamicModule[{var = False},
   Dynamic@Column[{
      PopupMenu[
       Dynamic[var], {False -> "one menu", 
        True -> "Add another Menu"}],
      If[var, #0[], ## &[]]}]] &[]