Breaking down very large Git repo into smaller repos and keeping history

Question

I got this ridiculously large Git repo in GitHub. This repo consists of several independent projects that I want to break down into their own separate repos, and continue their development there.

ridiculously-large.git
├───Project1
├───Project2
├───...
└───Project1000

For now, I'm interested in Project2. I want to "clone" Project2 only, along with its history, to a new repo Project2.git exclusively dedicated to it, and leave ridiculously-large.git only for historical purposes.

So far, I've been somewhat successful with the following approach:

1. First, create a brand new repo in GitHub, Project2 with no commit history (this seems to be important in order to avoid those "fatal: refusing to merge unrelated histories...") and clone it.

   git clone https://github.com/me/Project2.git
   

   Thus, my local repo is already configured with default remote origin pointing to the new GitHub repo that will be specific to further development of Project2.

2. Then, within the new local repo:

   git config core.sparseCheckout true
   echo Project2/ >> .git/info/sparse-checkout
   

3. Then, I add a temporary upstream remote temp pointing to ridiculously-large.git:

   git remote add -f temp https://github.com/me/ridiculously-large.git
   

   So far so good. There is still nothing to commit.

4. Then, pull the master branch from the temp remote into a new work local branch:

   git pull temp master:work
   

   (I do this because of personal practice: I never work in any local master branch, whether or not the there is an upstream branch configured for it.)

5. This successfully accomplishes the goal of pulling only the sources related to Project2 into the working copy. So far so good.

6. Now, after I push work to the newly dedicated remote origin:

   git checkout work
   git push --set-upstream origin work
   

   ... I see THE WHOLE SHEBANG pushed to the newly dedicated remote for Project2, miserably defeating the entire purpose of this endeavor.

The Question(s)

• What is the established/recommended approach to achieve my goal?
• Is it possible to do it as a variation of what I'm doing (or not doing correctly?)
• Are there any other Git constructs more appropriate altogether?

Thanks in advance for any enlightenment!