You've got this all mixed up! Unless you're deglazing to do a pan sauce, the steak should be the LAST thing cooked. It's the centerpiece of the dish and the most expensive part, and shouldn't have a chance to get cold while you cook the other parts. Saving cleaning on a single saucepan is not worth eating cold steak.
You MUST use a stainless or cast-iron for this. As others touch on, teflon is not safe for searing, and fast temperature changes will wreck it.
Here's how to do things:
- Julienne onions to saute, prepare and measure stock, trim fat from steak, cut or butterfly it if needed (this is your mise en place).
- Start preheating saute pan for onions
- Start butter melting in saucepan for roux
- Prepare roux and saute onions at same time, then add stock to roux, whisk, and season
- Transfer onions to bowl, cover to keep hot, and start to preheat pan for searing steak
- Cook steak while finishing sauce as needed, then rest it appropriately
- Plate steak, garnish, sauce, and serve
Alternately, here's another approach:
- Mise en place (prep ingredients, optionally pre-measure them)
- Prepare sauce in advance, set aside or keep hot. Or, even do it the day before and reheat it. It's a roux-based sauce; it won't go bad in the fridge, and will reheat beautifully.
- Saute onions. Or, caramelize onions beforehand, and just reheat them briefly and add.
- Cook steak
- Plate, sauce, serve
And, finally the deglazed pan sauce way
- Mise en place
- Sautee onions. Set aside in bowl (or on a serving plate)
- Cook steak
- Throw heavy cream or red wine in to pan, and deglaze by scraping vigorously with a utensil and stirring as it boils vigorously
- Quickly season this, and throw in the onions, a few pinches of parsley, pepper, salt, garlic, and paprika, plus optionally thyme
- Congrats, you now have a pan sauce with onions to throw on your steak
In a professional kitchen, you'd have your onions pre-sauteed or sauteeing at the same time as the steak, and your sauce will be prepared in bulk, beforehand. You'd only have to cook the steak and briefly heat or season the other two parts.