Pat the chicken breasts completely dry. Lay each one flat and slice it horizontally all the way through to make two thin cutlets, four cutlets in total. Each cutlet tapers, so lay them between two sheets of plastic wrap and give the thick shoulder end a few light taps with a mallet or the flat of a heavy pan until the whole cutlet is an even 1/2 inch. Season both sides with 1/4 teaspoon of the salt and 1/4 teaspoon of the pepper, and set them aside while you prep everything else.
Do the rest of your prep now, because once the pan is hot nothing waits. Grate the Parmesan finely and weigh out 33 g if you have a scale. Drain the sun-dried tomatoes and cut them into thin strips, mince the garlic, wash the spinach and dry it thoroughly (wet leaves water down the sauce), and measure the cream and the broth into separate cups. Put an instant-read thermometer where you can reach it: two steps below depend on it.
Heat the olive oil in a 12-inch skillet over medium-high until it shimmers, about 2 minutes.
Lay the cutlets in the pan in a single layer with a little space between them and leave them alone for 3 minutes, until the underside is deep golden. If they are touching, sear in two batches instead: crowded cutlets steam rather than brown. Flip and cook 2 to 3 minutes more, to 155-160°F (68-71°C) at the thickest point. Move them to a plate and tent loosely with foil; they finish cooking later, in the sauce. In a 10-inch pan, plan on two batches from the start and add about 5 minutes to the cook time.
Turn the heat down to medium. Add the butter and the sun-dried tomatoes and cook, stirring, for about 1 minute, until the butter has melted and the fat has taken on the red colour of the tomatoes.
Add the minced garlic and the dried oregano to the fat. Cook, stirring constantly, for just 30 to 60 seconds, until fragrant and the garlic is barely pale gold. This is the only moment in the recipe when either one gets cooked properly: garlic dropped into a finished cream sauce stays sharp and sulfurous, and dried oregano stirred into cream never dissolves. Do not let the garlic take on real colour or it turns bitter.
Pour in the chicken broth and scrape the bottom of the pan with a wooden spoon to lift every browned bit. Let it bubble 30 to 45 seconds, until the pan bottom is clean and the broth has cooked down to a thin film. Do not wait for it to visibly reduce by half; a quarter cup spread across a 12-inch pan is a wet smear with no depth to judge, and leaving it longer only scorches the fond you just lifted.
Turn the heat down to medium-low, or low if you have a strong burner. Pour in the cream and bring it to a bare simmer: small lazy bubbles at the very edge of the pan, never a rolling boil. Cook until the sauce thickly coats the back of a spoon and holds a clean line when you draw a finger through it, usually 1 to 2 minutes in a 12-inch skillet and 3 to 4 in a 10-inch. Go by the spoon, not the clock. Cream spread thin over a wide pan reduces fast, and pushing it past this point leaves it greasy and prone to splitting.
Keep the heat at medium-low and add the spinach in two handfuls, stirring the first one in until it collapses before the second goes in. Cook 1 to 2 minutes, until every leaf has wilted into the sauce. Two cups of spinach carry roughly a quarter cup of water, so the sauce will thin noticeably; that is normal. Simmer 1 to 2 minutes more, until it coats the spoon again. Fix a thin sauce here, before the cheese goes in, because you cannot simmer it down afterwards.
Turn the heat down to low. Return the cutlets to the pan along with every drop of juice that collected on the plate, spoon the sauce over them and warm through for 2 to 4 minutes, until the thickest cutlet registers 165°F (74°C), the USDA safe minimum for poultry. Do not let the sauce boil. Colour and clear juices are not reliable; use the thermometer. If it has not reached 165°F, cover the pan and give it another minute rather than turning the heat up.
Take the pan off the burner and switch the heat off. Wait 30 seconds; a heavy skillet holds enough residual heat to break the cheese even with the burner off. Now whisk in the Parmesan a small handful at a time, letting each addition melt smoothly before the next goes in. Season with the remaining 1/4 teaspoon salt and 1/4 teaspoon pepper, and taste. Salt goes in here, after the cheese, not before it. If the sauce has tightened too much, loosen it with a tablespoon of warm broth or cream.
Serve straight from the skillet, spooning the spinach and sauce over each portion.