B737-800 Captain reported a bird strike after take-off. Captain's airspeed became unreliable and both EECs were lost requiring a return to the departure airport.

Narrative

Shortly after take-off; climbing through approximately 600 ft. AGL we had a large bird strike. We lost the Captain's airspeed Indications as the bird had impacted the Captain's pitot tube and probably affected the air temperature probe; as we later on discovered loss of both engine Electronic Engine Controls (EEC's) as well. New hire First Officer (FO) was Pilot Flying (PF) with myself as Pilot Monitoring check pilot. This was the FO first PF leg. FO remained PF for the entire event. We continued the climb and flew into an overcast layer at 700 ft. AGL; climbing to approximately 2000 ft. MSL. We had the stick shaker; low speed aural alerts; loss of both flight directors. We quickly discovered the unreliable airspeed indicator. Completed some of the memory items associated with this issue. Told ATC we needed to return. We remained in IMC conditions for the remainder of the flight. Ran the checklist for unreliable airspeed. Overweight landing. Non routine landing; and subsequently ENG EEC alternate mode. We had an uneventful landing and taxi to the gate under our own power. Elected to use ATC vectors instead of holding. No injuries; no damage to aircraft other than pitot tube and possibly air temperature probe which affected the EEC's per maintenance call. Log book entries included. Bird strike with unreliable airspeed. Overweight landing. ENG EEC to alternate mode; and pulling voice recorder per company procedure. Received a phone call from ZZZ Approach Control where they filled out a bird strike paperwork via phone call with myself. Just want to say that our training works. This was the new hire FO first flying leg. Did an excellent job of maintaining aircraft control under these particular failures. The Flight Attendants also did a great job of taking care of passengers and preparing the cabin for an unplanned return to the airport.