Season the chicken on both sides with salt and pepper. In a 12-inch skillet over medium-high heat, combine milk, tarragon sprigs, garlic, butter, peppercorns and 2 teaspoons salt. Bring to a simmer, stirring, then reduce to low and submerge the chicken breasts in an even layer. Cover and cook, adjusting the heat to maintain a gentle simmer (only a few bubbles), until breasts are opaque throughout, 18 to 24 minutes, flipping halfway through.
Transfer chicken to a cutting board to rest. Pour poaching liquid through a fine mesh strainer set over a medium bowl; reserve 2 garlic cloves and 3 tablespoons strained poaching liquid, then discard the remaining solids and liquid.
In a food processor, combine tarragon leaves, parsley, capers, anchovies, vinegar, and reserved garlic and poaching liquid. Pulse until finely chopped, about 15 pulses. Add oil and process until combined, about 20 seconds.
Cut chicken breasts crosswise on the bias into ½-inch-thick slices and arrange on a platter. Spoon the sauce on top.
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