Set oven rack about 6 inches from the heat source and preheat the oven's broiler. Line a baking sheet with aluminum foil. Cut poblano in half from top to bottom; remove the stem, seeds, and ribs. Place cut side-down onto baking sheet.
Cook poblano pepper under the preheated broiler until the skin of the pepper has blackened and blistered, 5 to 8 minutes. Place in a bowl and tightly seal with plastic wrap. Allow pepper to steam as it cools, 10 to 20 minutes. Remove and discard skin; dice pepper.
Combine diced pepper, beans, bacon, onion, shallot, garlic, salt, and black pepper in a slow cooker. Pour in chicken stock.
Cook on High until beans begin to soften, about 3 hours. Taste broth to check seasoning. Stir in barbeque sauce and hot sauce. Continue cooking on High until beans soften and flavors are combined, about 3 hours more.