Return skillet to medium heat. Add 2 tablespoons unsalted butter to skillet and let butter melt completely, swirling and tilting pan occasionally to distribute butter across surface.
Once butter has melted, add 1 cup finely chopped yellow onion and stir to incorporate. Sauté onion approximately 8 to 12 minutes, stirring occasionally, until onions are softened and translucent.
Add 2 cloves garlic to skillet with softened onion and stir to incorporate. Sauté mixture 30 to 60 seconds or until garlic is just fragrant.
Once garlic is fragrant, pour ¼ cup dry white wine into skillet and stir to incorporate, making sure to scrape up any browned bits of food that may have stuck to surface of pan.
Increase heat under skillet to medium-high. Once liquid begins to boil, immediately reduce heat to medium-low and simmer mixture 2 to 3 minutes or until liquid has reduced by 50%.
Once liquid has reduced, remove skillet from heat and let cool slightly, approximately 60 seconds. While skillet cools, reduce heat on stovetop to low. After skillet has cooled approximately 60 seconds, add 2 teaspoons Dijon mustard and whisk to incorporate.
Once Dijon has been incorporated, slowly pour 1 cup heavy whipping cream (at room temperature) into skillet. Whisk gently to incorporate cream into other ingredients.
Return skillet to stovetop over low heat. Simmer sauce, stirring frequently, 3 to 5 minutes or until sauce is slightly thickened.
Once slightly thickened, season sauce with plenty of salt and freshly cracked black pepper, then add 2 tablespoons fresh lemon juice and 1 tablespoon chopped fresh tarragon and stir to incorporate.
Taste sauce and adjust Dijon mustard, salt, pepper, lemon juice, and/or tarragon as desired. Be sure to stir in any additions until fully incorporated.
When satisfied with flavor of sauce, return chicken breasts to skillet. Simmer chicken breasts in sauce while spooning sauce from skillet onto tops of chicken, 60 seconds. After 60 seconds, flip chicken breasts over and repeat, spooning sauce over chicken, until chicken breasts are warmed through.