Air carrier A321 pilot reported a loss of aircraft control during climb due to rapidly building thunderstorms; which resulted in flight attendant injuries due to turbulence. The flight continued to the destination airport where the flight attendants received medical attention.

Narrative

Enroute to ZZZ from ZZZ1 while climbing through FL230 we encountered mod+ turbulence navigating between a couple of cells 20 or so miles on either side as depicted on radar with a lower cloud deck between. The lower cloud deck seemed to have no development and zero radar returns. The seat belt sign was still on and Autopilot engaged. We were climbing at best rate to distance ourselves from the lower cloud deck and it appeared we would clear it by thousands of feet. As we got closer something seemed to be developing and; as we approached; I tried to bank right to avoid the very top of an incredibility fast building cloud. We inadvertently skimmed the top and side of this cloud and experienced the turbulence. The aircraft banked further right and then back to the left 30 degrees or so but remained on the Autopilot. The encounter was brief and only lasted seconds but it was enough for me to immediately call the cabin to check on the Flight Attendants (FAs). I assumed the worst when I didn't get a response after multiple attempts and sent a message via ACARS to Dispatcher about possible FA injury. When I finally made contact with #1 FA she said that #2 FA had fallen and was in pain but not sure of injury. As I gathered more information I told #1 to call operations and # 4 FA said he would (and did) call. I referenced the medical emergency flow chart throughout this process. On my numerous contacts with cabin crew and Dispatch I learned operations thought best course of action was to proceed to ZZZ and administer oxygen to #2 FA. I learned that FA #3 also had a bruised hand which I later found out was from #2 FA falling on to his hand by the jumpseat. Throughout the duration of the flight I continued to contact the cabin crew and Dispatch to confirm operation's recommendations remained. I talked to #2 FA and a deadheading ZZZ2 FA who said he would assume her duties. We got priority handling from ATC and requested medical assistance at the gate. We were met at the gate by a team of medical personnel and paramedics who walked #2 FA off the aircraft to a wheelchair for further evaluation. I asked and the paramedics evaluated the other FAs for potential injuries and they said they were ok to continue their trip.Cause: Incredibly fast vertical development of weather and closure rate. Also minimal bank angle of aircraft with Autopilot engaged.I keep thinking that turbulence may have been lessened by disengaging Autopilot and banking more aggressively but I am not sure if that would have worked with the closure rate and vertical development that I haven't seen before.