Rinse chicken inside and out; pat dry with paper towels. Loosen skin of breast and thighs and work some salt under skin. Rub remaining salt all over chicken and in cavity (see Notes). Chill, uncovered, at least 3 hours and up to overnight.
Preheat oven to 375°. Zest lemons. Slice 1 lemon; juice half of the other.
Pat chicken dry, inside and out. Rub zest under as much of the skin as possible and rub any remaining zest inside cavity. Rub chicken all over with 1 tbsp. olive oil and the thyme. Put lemon slices in cavity.
Set a V-shaped rack in a heavy roasting pan large enough to hold shallots. Put chicken in rack, breast side up. Add shallots to pan and drizzle with remaining 1 1/2 tbsp. olive oil, turning them to coat.
Roast chicken, basting every 30 minutes or so, until chicken leg moves easily and skin is brown and crisp, 1 1/2 to 1 3/4 hours (remove shallots after 1 hour and set aside). Tip chicken so juices from cavity pour into roasting pan. Transfer chicken to a carving board and let rest, covered with foil.
Meanwhile, make sauce: Pour pan drippings into a measuring cup with a pouring lip. Trim tops from shallots and squeeze soft insides into a blender. Pour off all but about 1 tbsp. fat from pan drippings and add drippings to blender. Add 1/2 cup broth and the wine and pulse until smooth.
Pour sauce into roasting pan. Cook, scraping up brown bits and adding more broth if you want a thinner sauce, over medium-high heat on your biggest burner (or straddling 2 burners) until sauce turns a nutty brown, about 10 minutes. Stir in 1 tbsp. reserved Meyer lemon juice, or more to taste. Pour sauce through a fine-mesh strainer into a serving bowl. Carve chicken, discarding lemon slices; serve with sauce.
Note: Nutritional analysis is per serving with 2 tbsp. gravy.