Slice fully thawed chicken breasts horizontally into 6 even cutlets about 1/2 inch thick.
Season cutlets with 1/2 teaspoon salt and pepper. Coat lightly in flour, shake off excess and discard contacted flour.
Heat oil in a wide skillet over medium-high. In two batches, sear cutlets 60 to 90 seconds per side until pale golden but still raw inside.
Cook shallot 2 minutes and garlic 30 seconds. Add 3/4 cup broth, scrape the pan and whisk in Dijon and thyme. Pour into slow cooker.
Arrange cutlets with minimal overlap. Cover and cook on low 2 to 3 hours, beginning checks at 1 hour 45 minutes.
Remove each cutlet when its thickest portion reaches 165°F. Rest on a clean plate for 5 minutes.
Whisk cornstarch with remaining cool broth. If sauce needs thickening, add slurry gradually and cook on high 5 to 10 minutes.
Switch off cooker. Add lemon juice and rinsed capers, taste and adjust salt. Whisk in cold butter, then zest and parsley; return chicken only to coat.