## Step 1: Define the optimization problem
The problem is to maximize the objective function $5.73x_0^2 + 1.63x_1$, where $x_0$ is the quantity of ravioli and $x_1$ is the quantity of rotisserie chickens, subject to various constraints.

## 2: List all the constraints
The constraints are:
- $x_0 \geq 0$ and $x_1 \geq 0$ (non-negativity)
- $x_0$ is an integer (ravioli must be a whole number)
- $4x_0 + 18x_1 \leq 100$ and $4x_0 + 18x_1 \leq 100$ (total healthiness rating)
- $19x_0 + 4x_1 \geq 67$ and $19x_0 + 4x_1 \leq 158$ (total grams of protein)
- $7x_0 + 25x_1 \geq 47$ and $7x_0 + 25x_1 \leq 119$ (total sourness index)
- $16x_0 + 23x_1 \geq 108$ and $16x_0 + 23x_1 \leq 197$ (total grams of fiber)
- $15x_0 + 7x_1 \geq 65$ and $15x_0 + 7x_1 \leq 149$ (total milligrams of iron)
- $-9x_0 + 2x_1 \geq 0$ (linear constraint)
- $4x_0^2 + 18x_1^2 \geq 57$ (quadratic constraint for healthiness rating)
- $x_0^2 + x_1^2$ does not have a direct constraint but is part of other constraints.

## 3: Convert constraints into Gurobi compatible format
We need to model this problem using Gurobi. Gurobi supports linear and quadratic constraints but requires integer variables to be defined explicitly.

## 4: Write the Gurobi code
```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define variables
    ravioli = model.addVar(name="ravioli", lb=0, type=gurobi.GRB.INTEGER)
    rotisserie_chickens = model.addVar(name="rotisserie_chickens", lb=0)

    # Objective function
    model.setObjective(5.73 * ravioli ** 2 + 1.63 * rotisserie_chickens, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(4 * ravioli + 18 * rotisserie_chickens <= 100, name="healthiness_rating")
    model.addConstr(19 * ravioli + 4 * rotisserie_chickens >= 67, name="protein_min")
    model.addConstr(19 * ravioli + 4 * rotisserie_chickens <= 158, name="protein_max")
    model.addConstr(7 * ravioli + 25 * rotisserie_chickens >= 47, name="sourness_index_min")
    model.addConstr(7 * ravioli + 25 * rotisserie_chickens <= 119, name="sourness_index_max")
    model.addConstr(16 * ravioli + 23 * rotisserie_chickens >= 108, name="fiber_min")
    model.addConstr(16 * ravioli + 23 * rotisserie_chickens <= 197, name="fiber_max")
    model.addConstr(15 * ravioli + 7 * rotisserie_chickens >= 65, name="iron_min")
    model.addConstr(15 * ravioli + 7 * rotisserie_chickens <= 149, name="iron_max")
    model.addConstr(-9 * ravioli + 2 * rotisserie_chickens >= 0, name="linear_constraint")
    model.addConstr(4 * ravioli ** 2 + 18 * rotisserie_chickens ** 2 >= 57, name="quadratic_constraint")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Ravioli: {ravioli.varValue}")
        print(f"Rotisserie Chickens: {rotisserie_chickens.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("No feasible solution exists.")
    else:
        print("Optimization failed.")

solve_optimization_problem()
```