I did just this (well, tried is more like it) 6 weeks ago, I have 1 fridge with a temp control thermostat that holds up to 4 cornies (I only have 2 and a fermenting bucket). In my case the California lager yeast (CL) (wyeast) wouldn't start fermenting until I cranked the temp up to 60ºF, so I let the traditional lager (saflager-23) ferment for a week at 50ºF (since most off flavors are produced during the first 2-3 days) and then took the temp up to 60ºF and the California lager started fermenting, kept it there for 2 weeks and then racked both onto their secondary with some gelatin finings and took the temp up to the high 70's for 2 days and then down to 35ºF'ish for the last 3 weeks, and they're still there aging and only getting better.
Partial mash (not steeping) and partial boil (I only have a 2.5 gallon pot and stove-top to work with). Don't have a scale so I use ounces by volume and sometimes adjust by sight (I'm an amateur cook as well, so that helps).
5-5.5'ish gallons came out of it
3# amber DME
3.3# light LME
12 oz (volume) munich malt
16 oz (volume) crystal 10L
Northern brewer hops @ 60' and 20'
Dumped onto 2 bags worth of ice (1 bag for each fermenter) and adjusted to SG 1.052 with RO treated water (I only use RO treated water, the water down here is harder than steel)
Both are medium body, have great head retention, slightly sweet (CL less so than TL), copper (TL) and amber (CL), could have used a third hop addition though. The California lager has cleared up better for some reason. My California lager has a spiced aroma as if I'd used cloves in the recipe. The TL has a sulfury tinge (both smell and taste, had to purge some of it with CO2 from forced carbonation) but that is to be expected in a TL.
Anyhow, this is my experience with 2 types of lager from the same batch of wort.