Heat a large skillet on medium heat. Add 3 tablespoons of butter, allow it to melt, and whisk it very frequently. It should melt, foam up a bit, and then start to smell nutty and delicious while darkening in color. (Browned butter very quickly turns to burnt butter so watch this carefully! If it does burn, dump it out and start with a new 3 tablespoons.)
Once the butter is browned, remove the skillet completely from the stovetop onto a heating pad on the counter.
Whisking constantly add in the fresh thyme (just pull off the leaves to equal 1 full tablespoon), oregano, garlic, paprika, and about 1/2 teaspoon salt + 1/4 teaspoon pepper. It will foam up. Whisk until completely combined and then add in all the squash. Stir until the squash is well coated in the butter.
Return the skillet with the squash to the stovetop and resume it at medium-high heat.
Spread the squash pieces out in an even layer and let cook, without stirring, so that they brown a bit on one side for about 4-5 minutes. Add in the green beans, a few extra springs of thyme, and then stir and spread the pieces out again. Let cook without stirring another 4-5 minutes.
Cover the pan and reduce the heat to low. Let the veggies cook for 7-15 minutes depending on the desired tenderness and size of squash (Mine is always done around 8 minutes). The squash should be fork tender and beans should be crisp tender.
Meanwhile, in a small bowl, toss together the flour, poultry seasoning (do not omit), and a generous amount of salt and pepper.
Slice 2 large breasts (that equal a pound) into half making 4 large and completely equal chicken pieces.
Dredge the pieces to use up the flour mixture.
In another large (12 inch) skillet, melt 2 tablespoons butter on medium-high heat.
When melted, add in the four breasts in an even layer.
(If the chicken browns too quickly, reduce heat.) Cook, turning as needed, (I cook 3-5 minutes on one side and then flip and cook for another 3-4 minutes on the other) until chicken is browned on both sides, is no longer pink in the center, and the juices run clear. (165 degrees F for a meat thermometer.)
Top the cooked veggies with the cooked meat.
Melt the last 1 tablespoon of butter in the microwave and stir it with some remaining thyme (~1 teaspoon) pour on top of the chicken and generously top the chicken and veggies with freshly grated Parmesan cheese.