Scikit learn provides you two approaches to linear regression:

1. LinearRegression object uses Ordinary Least Squares solver from scipy, as LR is one of two classifiers which have closed form solution. Despite the ML course - you can actually learn this model by just inverting and multiplicating some matrices.

2. SGDRegressor which is an implementation of stochastic gradient descent, very generic one where you can choose your penalty terms. To obtain linear regression you choose loss to be L2 and penalty also to none (linear regression) or L2 (Ridge regression)

There is no "typical gradient descent" because it is rarely used in practise. If you can decompose your loss function into additive terms, then stochastic approach is known to behave better (thus SGD) and if you can spare enough memory - OLS method is faster and easier (thus first solution).

What you want is not batch gradient descent, but stochastic gradient descent; batch learning means learning on the entire training set in one go, while what you describe is properly called minibatch learning. That's implemented in sklearn.linear_model.SGDClassifier, which fits a logistic regression model if you give it the option loss="log".

With SGDClassifier, like with LogisticRegression, there's no need to wrap the estimator in a OneVsRestClassifier -- both do one-vs-all training out of the box.

# you'll have to set a few other options to get good estimates,
# in particular n_iterations, but this should get you going
lr = SGDClassifier(loss="log")

Then, to train on minibatches, use the partial_fit method instead of fit. The first time around, you have to feed it a list of classes because not all classes may be present in each minibatch:

import numpy as np
classes = np.unique(["ham", "spam", "eggs"])

for xs, ys in minibatches:
    lr.partial_fit(xs, ys, classes=classes)

(Here, I'm passing classes for each minibatch, which isn't necessary but doesn't hurt either and makes the code shorter.)