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This is a reland of 86556051

The original CL missed a dep requirement in the Chrome branded
Windows build. This has been fixed:
https://chromium-review.googlesource.com/c/chromium/src/+/2543873/2/chrome/browser/win/conflicts/BUILD.gn

Original change's description:
> Extract browser_process.h/cc into own gn target
>
> This CL pulls out browser_process.h and browser_process.cc into their
> own gn target so that other gn targets can depend on it without creating
> cyclic dependencies with //chrome/browser:browser.
>
> This is required for CL:
> https://chromium-review.googlesource.com/c/chromium/src/+/2528519
>
> Bug: 1147410, 883570, 925153
> Change-Id: I17ef1f0b1683c4abb7588a6ef24ec89f15647545
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2529309
> Commit-Queue: Alan Cutter <alancutter@chromium.org>
> Reviewed-by: Filip Gorski <fgorski@chromium.org>
> Reviewed-by: Glen Robertson <glenrob@chromium.org>
> Auto-Submit: Alan Cutter <alancutter@chromium.org>
> Cr-Commit-Position: refs/heads/master@{#828077}

Bug: 1147410
Bug: 883570
Bug: 925153
Change-Id: I18faaddc3850ceafb924bda6d8e197f886e79433
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2543873Reviewed-by: Glen Robertson <glenrob@chromium.org>
Reviewed-by: Filip Gorski <fgorski@chromium.org>
Commit-Queue: Alan Cutter <alancutter@chromium.org>
Cr-Commit-Position: refs/heads/master@{#828442}