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In Virginia Woolf's novels, she used double em-dashes quite often. In Latin Modern, the two dashes closely set together without gap. However, In ebgaramond there is a little gap which is quite unpleasant.

I found online this will do the trick:

newcommand{dmd}{{{---kern-1pt---}penaltyexhyphenpenalty}}

However, it doesn't do it for me, my code is the following in LyX Preamble with document class of Book KOMA-SCRIPT:

usepackage{ebgaramond} 

usepackage[T1]{fontenc}

newcommand{dmd}{{{---kern-1pt---}penaltyexhyphenpenalty}}

The text in concern:

This one, thanks to the hospitality of the host, had lasted far into
the afternoon. The beautiful October day was fading and the leaves
were falling from the trees in the avenue as I walked through it. Gate
after gate seemed to close with gentle finality behind me. Innumerable
beadles were fitting innumerable keys into well-oiled locks; the
treasure-house was being made secure for another night. After the
avenue one comes out upon a road---I forget its name---which leads
you, if you take the right turning, along to Fernham. But there was
plenty of time. Dinner was not till half-past seven. One could almost
do without dinner after such a luncheon. It is strange how a scrap of
poetry works in the mind and makes the legs move in time to it along
the road. Those words------

This is called a horizontal bar, also known as a quotation dash. It is used to introduce quoted text (sometimes). This is the standard method of printing dialogue in some languages.

There is no support in the standard TeX fonts, but one can use instead as special macro, or just use an em dash.

newcommandhemdash{hbox{---}kern-.5em---}    

You can do this:

documentclass{article}

newcommandhemdash{hbox{---}kern-.5em---}

begin{document}



This one, thanks to the hospitality of the host, had lasted 

far into the afternoon. The beautiful October day was fading and 

the leaves were falling from the trees in the avenue as I walked 

through it. Gate after gate seemed to close with gentle finality 

behind me. Innumerable beadles were fitting innumerable keys into 

well-oiled locks; the treasure-house was being made secure for another 

night. After the avenue one comes out upon a road---I forget its 

name---which leads you, if you take the right turning, along to Fernham. 

But there was plenty of time. Dinner was not till half-past seven. 

One could almost do without dinner after such a luncheon. It is strange 

how a scrap of poetry works in the mind and makes the legs move in 

time to it along the road. Those wordshemdash





end{document}

In comments, you asked for a solution which does not involve modifying the content text, but only the preamble. This is indeed possible, if you change the backend for LyX to LuaTeX. I don't use LyX, but their website says that this is an option for versions later than LyX 2.0.
Then change the preamble to this:

usepackage{ebgaramond}

usepackage{luatextra}

begin{luacode*}

  local function translate(line)

      return string.gsub(line, "%-%-%-%-%-%-", "{---\kern-.1em---}\penalty\exhyphenpenalty")

  end

  luatexbase.add_to_callback("process_input_buffer", translate, "Translate")

end{luacode*}

Note that you need to remove the usepackage[T1]{fontenc} line.
We have added a Lua function that searches each line of TeX for ------ (each - must be escaped as %- in Lua patterns), and replaces it with the double em-dash command you found.
It should look like this:

This is derived from this answer.

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