To solve Amy's bakery problem using linear programming, we first need to define the decision variables and the constraints based on the given information.

Let:
- \(x_1\) be the number of croissants produced.
- \(x_2\) be the number of ficelles produced.

The objective is to maximize revenue. Given that each croissant can be sold for $4.5 and each ficelle for $3.5, the objective function can be written as:
\[ \text{Maximize} \quad 4.5x_1 + 3.5x_2 \]

There are two main constraints:
1. **Mixing Time Constraint**: Each croissant requires 12 minutes of mixing, and each ficelle requires 17 minutes. There are 350 minutes available.
\[ 12x_1 + 17x_2 \leq 350 \]

2. **Vanilla Extract Constraint**: Each croissant requires 2 tablespoons of vanilla extract, and each ficelle requires 1 tablespoon. There are 45 tablespoons available.
\[ 2x_1 + x_2 \leq 45 \]

Additionally, \(x_1 \geq 0\) and \(x_2 \geq 0\), since the number of bread items cannot be negative.

Now, let's write the Gurobi code in Python to solve this linear programming problem:

```python
from gurobipy import *

# Create a new model
model = Model("Bakery_Problem")

# Define the decision variables
x1 = model.addVar(vtype=GRB.CONTINUOUS, name="Croissants")
x2 = model.addVar(vtype=GRB.CONTINUOUS, name="Ficelles")

# Set the objective function
model.setObjective(4.5*x1 + 3.5*x2, GRB.MAXIMIZE)

# Add constraints
model.addConstr(12*x1 + 17*x2 <= 350, "Mixing_Time")
model.addConstr(2*x1 + x2 <= 45, "Vanilla_Extract")

# Optimize the model
model.optimize()

# Print the results
if model.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Croissants: {x1.x}")
    print(f"Ficelles: {x2.x}")
    print(f"Maximum Revenue: ${model.objVal:.2f}")
else:
    print("No optimal solution found")
```