The potato is probably not as important as the method. Simply frying them will not get you the fry you want. Basically you boil them, then double fry them. Here are the steps for The Perfect French Fry:
Ingredients
2 pounds russet potatoes (about 4 large), peeled and cut into
1/4-inch by 1/4-inch fries (keep
potatoes stored in a bowl of water)
2 tablespoons distilled white vinegar
Kosher salt
2 quarts peanut oil
Procedure
Place potatoes and vinegar in saucepan and add 2 quarts of water and
2 tablespoons of salt. Bring to a boil
over high heat. Boil for 10 minutes.
Potatoes should be fully tender, but
not falling apart. Drain and spread on
paper towel-lined rimmed baking sheet.
Allow to dry for five minutes.
Meanwhile, heat oil in 5-quart Dutch oven or large wok over high heat
to 400°F. Add 1/3 of fries to oil (oil
temperature should drop to around
360°F). Cook for 50 seconds, agitating
occasionally with wire mesh spider,
then remove to second paper-towel
lined rimmed baking sheet. Repeat with
remaining potatoes (working in two
more batches), allowing oil to return
to 400°F after each addition. Allow
potatoes to cool to room temperature,
about 30 minutes. Continue with step
3, or for best results, freeze
potatoes at least over night, or up to
2 months.
Return oil to 400°F over high heat. Fry half of potatoes until crisp and
light golden brown, about 3 1/2
minutes, adjusting heat to maintain at
around 360°F. Drain in a bowl lined
with paper towels and season
immediately with kosher salt. Cooked
fries can be kept hot and crisp on a
wire rack set on a sheet tray in a
200°F oven while second batch is
cooked. Serve immediately.
Worth reading is also the science behind the perfect fries.
It sounds to me like the issue may be that you're crowding the pan.
Basically, to get everything nice and brown and crispy, you need enough space for all of the steam to escape. That picture you showed has potatoes stacked on top of each other -- that means as the bottom items cook, they're going to end up steaming the items above them.
At a diner, they have a large griddle to work with -- they can really spread things out. You're not typically that lucky in a regular kitchen, as you don't have as much space, and you have a lip on the pans that'll hold the steam in.
So, either work in smaller batches, or consider recipes that use an oven -- using sheet pans instead of a pan on the stove solves much of the problem.
One other trick is that most diners don't start from raw potatoes -- maybe with hash browns, but not for home fries, you're not going to get the nice soft interior in a reasonable amount of time unless you start with a potato that's already been baked or boiled. (If you're doing things in the oven, you might be able to, but not in a pan)
Just for reference ... I have a 14" cast iron skillet that I use for home fries ... and it's about the right size for cooking a single large potato, which might be two servings, maybe three for kids. (I tend to cook carb-heavy meals).
update : I probably should've stated this directly -- you want the chunks of potato to form a single layer in the pan, with space in between them.
Best Answer
I wouldn't deep fry potatoes to par cook them for use with anything other than future frying. That cooking method gives the potatoes a hard exterior that won't cook as easily on your griddle.
Instead for home fries I prefer to blanch my potatoes in boiling water. This makes them soft and absorbent to seasoning and the starch is pulled to the surface which then gets that nice crispy exterior that you want at the END of the homefry cooking process.
Oh and since your size will effect your cooking time I will just give you a ballpark 7-10 minutes but really you are looking for about 60-75% doneness.