The tests look great, thank you!

Poco::Net::TCPServer already has the function stop(). It looks misleading to have both stop() and close() functions in TCPServer. Can we rename close() -> stop() and also call Poco::Net::TCPServer::stop() in this function?

So the only difference between TCPServer and Poco::Net::TCPServer is that TCPServer closes the socket. Can you please add a comment why Poco::Net::TCPServer::stop() is not enough and we also need to close the socker?

It seems it would be clearer to define a constant instead of using the magic number:

constexpr size_t connection_check_timeout = 1; /// 1 second
while (!in->poll(1000000 * connection_check_timeout))

We can probably make the code clearer if we add a new function named like updateServers(), and move the following code in there.

Sharing of a mutex doesn't look very nice to me. Maybe it's better to change something. For example, it seems instead of passing servers_to_start_before_tables_, servers_lock_, servers_ we can pass a function of type std::function<std::vector<ProtocolServerMetrics>()>, where ProtocolServerMetrics is defined as following

struct ProtocolServerMetrics
{
  String port_name;
  size_t current_threads;
};

Thus we will able to turn both the mutex and the lists of servers back to simple local variables in Server.cpp, without necessity to share them with another module.