Nested Stored Procedure Connections
A solution could be to build a simple SSIS package that runs the stored procedure you need; then call it from tsql in this way:
declare @execution_id bigint
 exec ssisdb.catalog.create_execution 
  @folder_name = 'folder'
 ,@project_name = 'project_name'
 ,@package_name = 'run_sp.dtsx'
 ,@execution_id = @execution_id output

 exec ssisdb.catalog.start_execution @execution_id

This call is asyncronous by default