Add ¼ cup all-purpose flour and ½ cup low-sodium chicken broth to small mixing bowl. Whisk ingredients together until fully combined, with no clumps of flour remaining. Set aside.
Place 1 pound boneless, skinless chicken breasts on cutting board or large plate. Sprinkle 1 tablespoon Everything but the Bagel seasoning onto chicken breasts, covering all sides thoroughly.
Place medium skillet on stovetop over medium heat. When pan is warm, add 1 tablespoon neutral cooking oil and continue heating until oil is hot and shimmery. Tilt pan as needed to distribute oil across entire surface.
When oil is hot, place seasoned chicken breasts in skillet. Cook chicken breasts 3 to 5 minutes or until opaque and lightly browned. Flip chicken breasts over and cook on other side 3 to 5 minutes, or until lightly browned and firm. Note: internal temperature of chicken breasts should read 160° Fahrenheit when checked with internal meat thermometer.
Transfer seared chicken breasts to cutting board and set aside until chicken is cool enough to handle. Once cooled, chop chicken breasts into bite-sized pieces and set aside.
Place large dutch oven on stovetop over medium heat. Add 1 tablespoon unsalted butter to dutch oven and continue heating until butter has melted.
When butter has melted, add white parts of green onions and 1 teaspoon minced fresh garlic to dutch oven. Stir to incorporate, then sauté ingredients until tender and fragrant, approximately 3 minutes.
Add 3 ½ cups low-sodium chicken broth and flour slurry to dutch oven. Whisk ingredients together until fully incorporated.
Remove dutch oven from heat and let cool slightly. Add 1 8-ounce block full-fat cream cheese to pot and whisk slowly until cream cheese begins to warm, then return dutch oven to heat. Continue whisking until cream cheese is fully incorporated into broth.
Once cream cheese has been incorporated, add chopped everything bagel chicken to dutch oven and stir to incorporate. Simmer chicken in soup mixture 2 to 3 minutes, then taste soup and add salt and freshly cracked black pepper as desired.
When satisfied with flavor of soup, add green parts of green onions to dutch oven and stir to incorporate.