How to hook up circuit to drive 5x7 1 Watt White LED
It seems you want to design a 5 digit 7 segment LED display without decimal points which are Multiplexed from a 12 bit parallel port and can handle 350mA per segment. Out of 5 thousand different LED driver designs that I searched none fit your requirements. Most now are serial input parallel output and all in SMT packages, which may not be suitable for your soldering or design experience.

You can choose either Common Anode (CA) or Common Cathode (CC) design in your circuit. This means you need a column driver that can handle all 7 LEDs in the matrix or 2.5A but at 5 times the average current since you chose a Mux'd matrix design of 5 digits, which means you need a digit driver capable of >12A from a logic level input.  I would try to find a 7 segment current limited driver solution then see use a high or low side switch that can handle 15A. The power source needs to be well filtered and isolated from your logic for this design current. The voltage needs to be just enough to drive both switches and the LED worst case with all segments on and no more to minimize switch loss temperature rise. After all you are implying a need to drive 35 LED's * 12W = 42 Watts and these designs can quickly become &lt;50% efficient if you are not careful and you don't want to put in massive heatsinks.

The 1.2W @350mA LED spec implies the LED voltage is 3.43V max. You need to regulate the current in each LED which tends to require about a 1 ~ 3V swing to be accurate in the current limiter depending on the design method, discrete (more complex but more efficient) vs IC.  The ideal solution would be an IC solution but the power dissipation is just too great for a small package. 

If you could live with 100mA current limit per segment your LED's will run cooler and 1 IC do the whole design. Unfortunately if not, it will take a lot of effort to design a circuit, buy parts, design a board and solder with SMT and manage your thermal issues.

If you can handle the software to drive a serial input shift register ( Clock and data) this chip can handle 8 segments up to 100mA for 1 digit.  Thus 5 digits with a common clock would need 6 parallel logic bits. But the elegance here is that each digit driver is latched, so you only need to send data on changes to the display and since it is not Mux'd you no longer need 5 High Side switches that each have to drive 12 Amps.

STP08DP05 Low voltage 8-bit constant current, LED sink with full outputs with error detection on a 5V supply.

The STP08DP05 is backward compatible in the functionality and footprint with STP8C/L596 and extends its functionality with open and short detection on the outputs.

This may require a bigger effort or simpler approach with 1/3W per LED.  If you can reduce your viewing angle, you can make up for the lost brightness with LED lenses to focus the beam.

Let me know which way you prefer Full power with big discrete 5x7 MUx 12A drivers or simple 5 serial digit drivers.  A full description of the overall task would be helpful too.