Brush lamb chops with oil and season with salt and pepper; sear well on both sides in a hot skillet, 1 to 2 minutes per side. Place chops on a full-size sheet pan. Roast at 350°F in convection oven (or 400ºF in conventional oven) for 5 to 6 minutes to an internal temperature of 140ºF, or cover and chill until ready to use.
Add remaining oil to skillet; heat over medium-high heat. Add shallots and garlic; sauté 1 to 2 minutes. Deglaze the pan with vinegar, figs and honey; simmer 1 to 2 minutes. Add stock; bring to a boil and reduce sauce by half, stirring occasionally. Just before serving swirl in butter pieces and parsley.
To serve: For each serving, plate one lamb chop over the top of 1 cup of greens; ladle with 1/3 cup of sauce. Garnish with 1/4 cup tomato halves and 1/4 cup of croutons if desired.