Heat oven to 500ºF with rack set in middle position. Brush a splash of oil on a sheet pan to coat.
Quarter 1 head cabbage through core and cut each quarter into 2 wedges, leaving core intact. Arrange wedges, flat side down, on prepared sheet pan. Brush 1 tablespoon oil on exposed cut sides of wedges and sprinkle with ½ teaspoon kosher salt. Flip wedges so oiled sides are flush with sheet pan. Brush second cut sides with remaining 1 tablespoon oil and sprinkle with ½ teaspoon salt and ¼ teaspoon pepper. Cover sheet pan tightly with aluminum foil and roast for 10 minutes.
Meanwhile, add ½ minced shallot to a small bowl; cover in cold water and set aside.
Remove sheet pan from oven and carefully remove foil, opening away from you to allow steam to escape out.
Return sheet pan to oven and cook until cabbage begins to brown on underside, about 10 minutes.
Remove sheet pan from oven. Using tongs and a thin metal spatula, flip each cabbage wedge. Scatter 12 ounces sliced kielbasa around cabbage then return to oven. Roast until edges of cabbage are browned and some leaves have crisped, another 10 minutes.
Remove sheet pan from oven and top with 1 can drained beans. Return sheet pan to oven and allow beans to warm through while preparing herb dressing (if you’ve already made the dressing, keep sheet pan in oven for 5 minutes before topping with dressing and serving).
Meanwhile, drain minced shallot; add shallot to a liquid measure cup. Add ⅓ cup chopped herbs, ¼ cup olive oil, 3 tablespoons vinegar, 1 tablespoon whole-grain mustard, 2 teaspoons Dijon, ½ teaspoon sugar, and ¼ teaspoon pepper; whisk to combine then season to taste with salt.
Remove sheet pan from oven and spoon herb mixture in dollops over top; serve immediately.