Rub -- In a small bowl, add the olive oil, vinegar and seasonings and mix well. Rub this over the pork medallions and place in a small pan and cover well with saran wrap. You can also put them in a ziploc baggie if you prefer. Let them marinate at least 2 hours up to all day.
Pork -- Heat up a large skillet to medium high heat and add the butter and olive oil. Cook the pork 4-5 minutes on the first side and flip. It should only take 3-4 minutes on the second side. You want a nice sear on both sides and they should be medium, just a hint of pink. But cook them to the temperature you prefer. 140-145 is medium. Just don't overcook them. Remove the pork and set on a plate to the side and cover. And, remember, the pork will continue to cook a bit.
Sauce -- now in the pan that you cooked the pork in (still on medium to medium/high heat) add the shallots and stir until they start to soften, just a minute. Then deglaze with the white wine and cook another minute. Now, add the blackberry jelly, balsamic vinegar, and chicken broth, herbs, salt and pepper and let reduce. Simmer on medium low for about 3-5 minutes as the sauce reduces. Add in the butter right at the end to make a nice glaze.
Serve -- Plate the medallions over some cooked rice with the blackberry glaze. ENJOY!
NOTE: If possible, I would definitely cook the rice in chicken broth and then add some fresh thyme or rosemary to tie together all the flavors, and to add some depth to the dish.